The Northern Ireland global is set to join on a six-month loan, until his contract expires at Southampton.
Rangers are only behind leaders Celtic on goal difference as they seek their first Premiership title in eight years.
Jermain Defoe made his England debut in 2004 and has made 57 appearances, scoring 20 goals, including three appearances and a goal at the 2010 FIFA World Cup.
Locked level on points with Celtic, Rangers are in with a real chance and Gerrard described Defoe's signing as "a great addition to our dressing room".
It was the Light Blues' first league win over their Old Firm rivals since 2012 and their first over Hoops boss Brendan Rodgers in 13 attempts since the Northern Irishman took over at Parkhead in 2016.
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Dundee have confirmed Glen Kamara will join Rangers after the midfielder signed a pre-contract agreement with the Ibrox side.
Following the threats to Beaton, SFA chief Maxwell said in a statement: "It is with deep dismay that, in the last 24 hours, we learned of one of our match officials having had threats made against him and his family".
"He has done a lot for this club, and there we find a good solution where we have a win-win situation for both sides".
"There have been ups and downs, but since the day I arrived I have given my all to play my part in pushing this club forward".
Now surplus to requirements with Eddie Howe preferring Callum Wilson and Josh King up front, Defoe has been allowed to leave the club on loan and join up with former England teammate Steven Gerrard at Ibrox.