Burnley have signed England U21 international central defender Michael Keane from Manchester United on loan.
Republic of Ireland international Forde was named the Lions' player of the year last season but his current deal is due to expire this summer and he has previously been linked with a move to Celtic.
But chief executive Andy Ambler expects to hear back from Forde and winger Lee Martin, who has also been offered a new contract, over the next few days.
"As with any renegotiation things take time," Ambler told the club website.
"Fordey is out in the States at the moment with Ireland and Lee we've made an offer to, and he's away as well. I'm hoping that we can have some clarity with that over the next four or five days."
Ambler also revealed Millwall are waiting for an answer from out-of-contract Yeovil defender Byron Webster over a move to The Den, while the club have turned down a bid for centre-half Mark Beevers.