The Finnish international signed a pre-contract agreement with the Ibrox side earlier this month.
Dens Park boss Jim McIntyre and Rangers were keen to push the deal through in January. It was reported that Dundee were offered £50,000 for the player earlier this week.
However, reports suggest Rangers have paid a six-figure sum for the midfielder.
Kamara has not featured for Dundee since agree the deal with Rangers, at the same time as Kilmarnock winger Jordan Jones.
He has played 18 times this season but is yet to reach the heights of last season, following his arrival in Dundee from Arsenal.
Once the deal is completed he will become Rangers fifth January arrival after Jermain Defoe, Steven Davis, Matt Polster and Andy Firth.