Oyston is hopeful the 25-day search for Michael Appleton's successor is coming to a close after interviewing an unnamed manager on Monday, with reports strongly linking former Derby and Nottingham Forest boss Billy Davies to the post.
Thompson is currently in caretaker charge of the Seasiders - a role he also performed earlier this season when Ian Holloway left to take charge of Crystal Palace.
"We've interviewed a manager," Oyston said. "It was very encouraging and I feel we could work with him.
"We now have a 24-hour cooling-off period for both parties to think about things, and we'll see where it goes from there.
"All the managers (we've spoken to) have wanted to bring their own people in - one or two wanted to bring a team with them.
"That would have meant dismissing our current staff and that's not something we'd like to do.
"We'd like to keep Thommo here, but I suppose that's down to the new manager and Steve and how they work together.
"In an ideal world, we'll appoint a manager who would allow Steve to stay as assistant manager."