Lukaku scored 16 goals as the Toffees finished fifth to qualify for Europe for the first time in five years.
Osman, 33, is desperate for the Belgium international to be back at Goodison Park in time for his testimonial game against Porto on Sunday, 3 August.
"It would be good to have Romelu back - we know the quality he's got," he said.
"There might be a bit more to do to get him back here but it would be great if he could come back in time to play in the game."
Everton's only signing of the summer is Gareth Barry, who also spent last season on loan with the club, although they are close to completing a deal for Ferencvaros midfielder Muhamed Besic.
Osman hopes manager Roberto Martinez can make further signings.
"We try not to be distracted by speculation, but sometimes you can't help but hear the rumours about who the manager wants to bring in to the club," he said.
"Besic should be completed in the next few days and we're aware of the rumours about a couple more coming in. It would be good if any new arrivals could play in my testimonial."

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Inst led 6-3 at the break thanks to two Hume penalties while John Crowther slotted over a Campbell penalty.
Campbell piled on the pressure in the second half but could not find a way through a resolute RBAI defence, even when Mark Keane was yellow-carded.
Hume sprinted clear from deep in his own half to run in a try, which he converted in the game's final action.
Campbell will rue missed opportunities during the St Patrick's Day decider in a sun-kissed Belfast.
They dominated the latter stages of the first half and entire second period.
Penalty opportunities were turned down in favour of going for a try but they were thwarted by handling errors and Inst defending.
Campbell launched a final attack but Hume collected from a bouncing ball and raced away to seal a second straight Schools' Cup triumph.
RBAI captain Conor Field said: "I can't describe my feelings after that but I am so proud of the boys.
"Credit must go to Campbell as they put us under so much pressure although we defended strongly throughout the game.
"To see James run clear to win it for us so late in the match was absolutely brilliant."
Everton midfielder Leon Osman hopes striker Romelu Lukaku will return to the club after impressing on a season-long loan from Chelsea.
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James Hume scored a late try as RBAI beat Campbell College 13-3 at Kingspan Stadium to retain the Schools' Cup.