The St Mary's have been scotuing Stuart Armstrong and Moussa Dembele but Rogic's performances for Celtic have also caught the attention of recruitment chief Ross Wilson, who watched the Australian dominate Rangers in the 5-0 win.
The Liverpool legend is now an academy coach at Anfield but it emerged last week that Rangers were interested in making the 37-year-old their next permanent manager.
The Daily Record says Gerrard had agreed to take over at Ibrox from the start of next season.
Today's Hoops rumour mill: Southampton linked with move for Australian star and ex-Celtic boss warns Steven Gerrard.
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He told BT Sport: "I have held initial talks and I have plans to pick them up in a couple of days" time.
And Houllier has given Gerrard some advice if he is to become the new Rangers manager.
Former Liverpool player and Celtic boss John Barnes told Sky Sports Gerrard will be "going in at the deep end" if he takes the Rangers job.
After retiring from football in 2016 after his spell in the MLS with LA Galaxy, the Reds legend has been coaching within the Premier League side's development ranks while earning his UEFA coaching badges.
Rangers are now third in the Scottish Premiership, 13 points behind Celtic, who clinched a seventh straight title when they defeated their Glasgow rivals at the weekend.