The Canadiens initially offered Markov a one-year contract, but the source indicated that they are willing to go to two-years.
Markov, 35, would prefer a three- or four-year agreement. The two sides continue to negotiate.
The Habs are also planning on discussing a contract extension with Gionta. Canadiens GM Marc Bergevin has explored the trade market on Gionta, but a move is unlikely.
Gionta's agent, Steve Bartlett, told TFP on Tuesday morning that he "would expect to discuss" a new contract after the deadline passes.
Bartlett indicated that "we haven't really talked too much about it as yet," but added that he is "open to discuss."
Markov and Gionta can become unrestricted free agents July 1, if unsigned.