O’Neill is currently in charge of the Northern Ireland national football team, who have just failed to qualify for the 2018 World Cup finals in Russia next summer.
The Gers are on the hunt for a new permanent manager following the departure of Pedro Caixinha in October.
Caixinha parted company with the Ibrox outfit due to a run of disappointing results in the Scottish Premiership.
The Scottish Sun has also claimed that Sunderland in the Championship in England are also interested in O’Neill.
Everton are on the lookout for a new permanent manager as well, with David Unsworth currently in charge of the first team following the dismissal of Ronald Koeman.
Former Northern Ireland international midfielder McIlroy has suggested why O’Neill could take the Rangers managerial job.
“He has big, big questions to ask himself,” McIlroy told The Herald about O'Neill. “If Scotland were interested he would have to ask himself: ‘Why would I leave Northern Ireland for Scotland?’
“Is it because there are far more players to work with and they are renowned as a bigger nation?
“He has to ask himself does he still want that sort of role or does he crave day to day involvement at a big club. Would that be appealing to him?
“He’s never had a taste of life as a manager at a major football club. If a big club, a Glasgow Rangers or an Everton, comes in he may decide that is his next challenge.”