Owen Coyle has revealed the shot-stopper, who spent much of last season as back-up to Adam Bogdan, has yet to decide whether his future lies with the relegated club
The former Finland international has spent 15 seasons at the Lancashire club and was awarded a testimonial match in 2009 to mark his 12th year at the club.
However, after picking up an injury at the turn of the year the 37-year-old lost his place in the starting XI to Adam Bogdan, but despite not being first choice his manager is keen to keep him.
"I want to make it clear that I want him [Jaaskelainen] to stay," Coyle told reporters.
"We know we have Adam, and that would give us two top keepers, but we want that strength in depth.
"I had a very good chat with him last week and he knows my feelings on the matter.
“We know how good a goalkeeper Jussi is, the fantastic service he's put in for Bolton Wanderers over a number of years, and we have put an offer to him that we think is a good one."
Should Jaaskelainen choose to stay at the Reebok Stadium, he could face an uphill challenge to regain his place in the starting XI.
Not only did Bogdan end the previous campaign as first-choice, the Hungarian goalkeeper is also 14 years his junior and Coyle was full of praise for Bogdan’s contribution in what was, ultimately, a disappointing season.
He continued: "Adam finished last season as one of the top few keepers in the Premier League for me.
"Jussi was disappointed when he lost his spot, and he was unlucky that it happened because of an injury but he knows how far Adam has come along.
"It's up to him now whether he wants to stay and challenge him, or move on."