The midfielder, sidelined for three months because of a knee injury, managed to come through against during a 75-minute run-out against Championship club Doncaster Rovers on Tuesday, helping Hull celebrate a 5-0 behind closed-doors win.
Return bid: Bullard hopes to be fit to face Everton
This followed a 55-minute appearance in a reserve match last week and Tigers manager Phil Brown said: 'Jimmy is fine but we'll have to see how he is for the rest of the week.
'I'm not making any promises that he's going to start on Sunday but if he's somewhere near, I'm sure he'll be kicking my door down to make sure he's involved.
'The motivation will be there for everyone to be involved this weekend and Jimmy will be no different.'
Bullard's last return from injury saw Hull embark on a four-game unbeaten run and such a statistic will be on Brown's mind as he prepares for Sunday's showdown.
He added: 'Jimmy is the kind of problem that we all want as a manager.
'It's a decision I've still got to make. I've got an idea how I want to play this weekend so we'll piece it all together in the days building up to Everton.'
FA set to investigate former Hull City chairman Paul DuffenGeovanni in line for Hull return: Tigers midfielder expected back on the training ground next weekHULL CITY FC