Under Civ 5 Gods and Kings, you can't engage in a Research Agreement unless you have a Declaration of Friendship with the other nation. However, I can't find a way to propose a DoF.
Is it possible to propose a Declaration of Friendship? Or is it something only the other nation can instigate?
(I also can't immediately find how to Denounce someone, but that's probably a different matter).
Best Answer
Both of these options (Declaration of Friendship and Denouncing) are available under the Discuss option when you first open the diplomacy screen to another nation.
I do believe you can't perform either if you're recently performed that action (so you can't propose friendship to a nation who recently rejected a previous proposal), or if the intended state is currently active (you can't propose friendship to your friends).
However, it is apparently possible to denounce friends or befriend those you've denounced, but the first is likely to cause you diplomatic issues while the latter is very unlikely to be accepted.
Finally, these things expire over time, so when that happens you'll be able to suggest a new declaration / denounce them again.