The midfielder is set to snub interest from Manchester City and Tottenham Hotspur in favour of Ibrox after meeting with Rangers boss Mark Warburton.
The 19-year-old is out of contract this summer and would be free to move to Glasgow having stalled on negotiating a new deal with Liverpool due to a lack of first-team opportunities.
It is thought a development fee would still have to be paid to his former club, as Liverpool made an attempt to sign the player to a new deal, and this will be in the range of £250,000.
Rossiter is a tenacious and tough centre midfielder with a decent touch and good eye for a pass. He has made five appearances for the first-team over the past two seasons, including one English Premier League fixture against Arsenal.