Allardyce is the favourite to succeed Koeman at Everton.
Koeman was sacked after a poor start to his second season in charge of Everton, who are still searching for the Dutchman's successor more than a month later.
Rooney returned to his boyhood club earlier this year, with Romelu Lukaku travelling in the opposite direction to Manchester United.
But Rooney has so far struggled fill the void left by the Belgian striker, who scored 26 times for Everton last season.
Wright, though, has tipped England's record goalscorer to rediscover his form if current favourite Sam Allardyce takes over at Goodison Park.
"He’s the kind of manager that Rooney will probably respond to," Wright told 5 live's Football Daily on Monday. "I’m not sure how he got on with Koeman. I’m not sure if it was Koeman who bought him or Bill Kenwright wanted that romantic signing back. I think that Sam Allardyce is that kind of manager that would probably utilise him to the best of his ability, at this point. Because he’s a manager he probably does respect.
"Whatever you say about Sam, with him there you’d fancy Everton to be saved. So you’d have to say he’s the one who can get the Bunsen burner under Rooney."
Allardyce's appointment is far from cut and dry at this stage, with O Jogo reporting on Monday that Paulo Fonseca is 'close' to being hired.
Fonseca led Shakhtar Donetsk to the Ukrainian title last season, and has them on the verge of qualification to the Champions League's last 16.