Fletcher was suspended for Saturday's 1-1 Scottish Premier League draw with Motherwell but was still the focus of attention after the game.
After seeing his side collect a point at home, Hibs boss Paatelainen said: "I am confident he will stay here.
"I don't think it's difficult for him to be honest (that the bid was turned down); from a players' point of view, he realises he is doing things right.
"Eventually in football, there is a time when a player moves on. I don't feel the time is right yet.
"I have had a chat with him and told him I don't think it's the right time.
"It's not entirely in my hands because you have to take things like finances into account, too, but we are in a position that we don't have to sell anyone."
In Fletcher's absence, Derek Riordan gave Hibs the lead in the eighth minute after finding the net from Ross Chisholm's cutback.
However, Motherwell rescued a point in the 75th minute when David Clarkson converted a penalty following Steven Thicot's foul on Jamie Murphy.
Hibs midfielder Dean Shiels looked on from the directors' box following Doncaster's announcement he was set to join them.
Asked what the situation was regarding the Northern Ireland international, Paatelainen said: "He is still our player; there is a risk that we lose him because he is out of contract this summer."
Meanwhile, Motherwell manager Mark McGhee insists he was always optimistic his side would take something from the game after seeing his players cause problems after half-time.
He said: "I thought we were going to score; I thought we were going to score earlier to be honest.
"We found an energy in the second half that was missing in the opening period. We got the goal but it just didn't come soon enough for us to get a winner."
McGhee insists his side's slow start to the match was not down to the fact they were missing Chris Porter after the striker agreed to join Derby.
He added: "I don't think the boys were thinking about that too much but, as a manager, you wished he was out there because we missed his ability and strength; he had been playing really well recently."