In attempting to answer this question, I use the definition of the covenant I wrote of in the past. A covenant is a formalization of a personal relationship. This leads me to answer both “yes” and “no.” It depends on how you approach the Adamic administration. If you look at the creation of Adam in terms of paternity and sonship, the tendency is to say “no.” If you look at the creation of Adam in terms of Creator and creature, the tendency is to say “yes.”
This question would be very easy to answer if the Adamic administration was referred to as a covenant in scripture. It is not. There is the possibility that Hosea 6:7 refers to a covenant with Adam. It may also refer to a more general covenant with mankind, such as the Noahic covenant. It is more responsible to prove that the Adamic administration fits the concept of the word covenant before we argue for one interpretation or the other in Hosea 6:7.
I want to argue first that the Adamic administration is revealed as a father-son relationship. Though this relationship is covenant-like, it is not necessarily a covenant. For this, I use the arguments of Jason Van Vliet in his graduate work.
He draws from a number of places in scripture to prove that the image of God is revealed in Adam’s sonship to God. Luke is most explicit. In the genealogy that Luke gives for Christ at the end of Luke three, Luke refers to Adam as a son of God, just as Seth was the son of Adam.
Where did Luke get this from? It is likely that Luke got this from his understanding of the image of God in Genesis 1. God makes Adam in his image. When the genealogy of Adam is given in Genesis 5, we are told that Seth is made in the image and likeness of Adam. The image of God seems to be about a father-son relationship.
Is a father-son relationship a covenant? I’ve already suggested that it is hard to call it a covenant. I believe that an adoption can be properly termed a covenant. Adoption is a legal process, which allows two individuals to act as Father and son. This is what happens in the covenant made after sin. By Christ, God worked it so that we may have the relationship of son and father that Adam had lost. There is no formalization to the relationship of a natural-born son to his natural father, however. If it is a covenant, it is one that springs from the way things are and does not need a legal creation. We could employ the distinction between nature and culture here. The father-son relationship springs from nature. Covenants are cultural, they build on natural bonds.
Compare it to marriage. In marriage, there is a creation of a new type of relationship, which the two type of individuals did not have before. In the case of a son or daughter, there is not a moment of the son’s existence, where he does not relate as a son to his father. One is natural. One is cultural. The father-son relationship is covenant-like. Marriage is a covenant.
Father-son relationships are covenant-like in that the relationship can be broken. The father or the son may forget their natural duties toward one another and betray one another’s trust. The father is called to rule well and the son to obedience and submission, at least in his growing years. Again, there is growth in that relationship, but there are also duties according to what we might call the created order. Thinking about Adam and God in this sense would suggest that covenant is not the best way to describe their relationship.
However, we must understand that this is an analogy. This is obviously true. God is God. Adam is a man. Adam is in the image of God. The children of Adam may even be referred to as gods, as they are in Psalm 82, but that means that they share the character of God, not the substance of God. This means that there is another legitimate way to think about the covenantal character of the creation of Adam.
When we begin with Creator God rather than Father God, a dramatic distance opens between God and Adam. God is eternal, infallible and unfailingly holy. Adam is none of those things. God grants Adam the image of God. God is intentionally creating a relationship between himself and Adam, which is formalized by sharing his image with Adam. From this perspective there is a covenant between God and Adam.
It is natural, or informal, in the sense that it springs from God’s imprint of his image upon Adam. But it is also formal in the sense that God chose that this should be the nature of his relationship with Adam. It is formal in the sense that God ordained that Adam should be in his image. When men have children they do not choose to have children in their image.
It is ultimately because God is not bound by his own created order, that we can understand his relationship with Adam as a covenant.
How should we speak of the Adamic administration then? Is it a covenant or is it not? It depends. Those theologians who wish to speak in Biblical terms as much as possible will be suspicious of calling it a covenant. I would count myself as one of those. However, it is important to recognize that there is a legitimacy in calling it a covenant as well, according to our philosophical understanding of what is happening in scripture. Such a way of speaking is not anti-scriptural. My preference is that covenant would not be the primary category for speaking of God’s relationship with Adam. Instead, we should think of that administration in terms of a father and a son. That is how God chose to teach us about his creation of Adam.
n.b. I’m not sure if the nature and culture distinction I mentioned works that well. The problem is Marriage is not merely cultural, it has a grounding in nature. If culture builds on and is rooted in nature, however, that is not a huge problem. The problem is with the popular understanding of culture today. We see culture as added to nature, not grounded in nature.