I'm having this issue also. An AI club has made a loan offer for a player i am interested in also, yet i cant seem to make an offer as well. I have room for 3 more non eu players in my team and still allowed 1 more player on loan. The club that bid has drastically lower rep than me so cant see it being him being interested in joining them over myself.
Edit: tested on a player who wasnt interested in joining me and despite this, i did have the "loan offer" tab instead of just "transfer".
I've also seen another player i was interested get loaned out before i was even allowed to make a counter bid. Basically, i hit continue to go to a new day and the following day, 3 messages came up saying a club mad an offer, had the offer accepted, then the player agreeing to go on loan and i couldnt do a thing about it, yet prior to this, the AI team wouldnt allow him to go on loan.
Clearly must be some kind of bug.
Edit: Nvm, it counts as 5 loans per season in total, not 5 loan players at the club at any time. Still stranger on certain players i dont even have the "loan" tab and on some i do