Manager Mark Warburton said: "He is seeing a specialist today in London to get a more detailed report.
"There is no surgery required, which is most pleasing.
"Hopefully he will be back for the end of the season, but there is a possibility he won't. It depends how his recovery progresses."
Waghorn's injury has led to Warburton calling for an end to the use of artificial surfaces in the Scottish Premiership and suggesting that the striker's "lacerations and immediate bruising" would not have been caused on a grass pitch.
The 26-year-old Englishman went off after winning and converting an early penalty in Rangers' 2-1 Scottish Cup replay win over Kilmarnock on Tuesday.
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And the club's top scorer wrote on Twitter that he expects to be out of action for six to eight weeks.
Rangers, who initially predicted that the striker would be missing "for a number of weeks", had added: "His rehabilitation will be expertly managed by the medical team and we look forward to welcoming him back sooner rather than later."
Waghorn, signed from Wigan in the summer, has scored 28 goals for Warburton's side, who lead the Championship table.
Nicky Clark replaced him at Rugby Park and scored Rangers' late winner to set up a home quarter-final against either Dundee or Dumbarton.
"His rehabilitation will be expertly managed by the medical team and we look forward to welcoming him back sooner rather than later," the club statement added.
Top scorer Martyn Waghorn could miss the rest of the season with the knee injury the Rangers striker picked up in midweek against Kilmarnock.