Ok, I think I've found out how to do this.
If you go into the individual player screen. Then into transfer. Then into Offer Via Transfer Room. Set your requirements. Press Offer Via Transfer Room. 90% of the time, I get no offers.
But, if you select Targets, then Add, then All Clubs From Nation. Then manually select Nations / Teams. Then Offer From Transfer Room, 75% of the time, I do get offers.
So then I thought I'd try the same thing with selling a player. Yep, you get more offers by manually selecting the Nations / Teams.