No. According to the Supreme Court, only the President is authorized to actually negotiate with foreign leaders. The Senate may advise him and ultimately must approve any proposed treaty, but they may only negotiate it through the President.
But did you know that Nixon did this while he was running for president (bargaining with South Vietnam), and Lyndon Johnson found out the night before the election, but couldn't reveal it because his source was the NSA, and therefore classified?
Committing a felony, and protecting the felons - we're quite good at that!