Yes, I transferred quite a few from GoDaddy to NameCheap, several close to the expiration date. They explained that even if it was close to the renewal date, if you had already paid for the transfer prior to expiration, it should be fine. Note that they are having another transfer discount sale next week, announcing up to 50% off.
Note, though, that it may best to try to do the transfer perhaps at least a week or so prior. That way, you don't have to worry about any delays due to authorization codes or Registrar release (which can take up to 7 days, at least with one registrar I have).