Douglas, 33, has made 16 appearances for the Bhoys this season, conceding just nine goals in the process.
Levein said: "We have made the offer to Douglas and it is now up to him.
Levein has also indicated he’s looking to tie up a deal with utility player Mo Sylla.
"The situation regarding Sylla is a bit different because he is still under contract and he is bound to have other offers to consider, but I am hopeful we can do something there.
"He can play in a number of different positions and that is what we are going to need next season."