Party leader Andrew RT Davies told the Sunday Politics Wales programme the Tuition Fee Grant is unaffordable .
He said people who can afford it "should pay their share".
Welsh students in the UK currently pay the first Â£3,500 of fees with the Welsh government covering the rest.
The cost of the scheme would be Â£3.6bn over the course of the next assembly, Mr Davies said, insisting the money should be spent on the NHS and further education (FE) colleges.
He said: "We don't think it can continue. We've always said that and it's coming home to roost now.
"When you look at the cuts in FE and apprenticeships, and more importantly the devastating cuts to the NHS budget, when people can afford to pay for things they should be paying their share."
Fears have previously been raised about the cost of the scheme, as well as concerns by universities that they are underfunded compared to their counterparts in England.
The Welsh government has always said its funding protected Welsh students after the UK government decided to raise tuition fees in England.
The funding arrangements of universities in Wales, including the Tuition Fee Grant, are currently being reviewed by the Welsh government, but it is not due to publish its report until after next year's assembly election.
The Welsh Conservatives will scrap the grant which pays the majority of tuition fees for Welsh students, if they are in government after next year's assembly election.