Francis said: "Clarke is going to be out for the rest of the season.
"He has seen a specialist and we will now decide when and where he has his operation."
Clarke's understudy, Alex Kolinko was forced to play in pain for the second half of Wednesday's goalless draw at Sheffield Wednesday after picking up a groin injury.
So, if Kolinko fails to recover in time, Francis may be forced into bringing in another keeper ahead of the clash with Gillingham on Sunday or take a chance with 16-year-old substitute keeper Lance Cronin.