Fred has fallen out with the French giants after coming back late off a winter break to his homeland, on top of being in and out of their side this season.
He has been hotly tipped for a move to Paris St Germain and Lyon president Jean-Michel Aulas said: "If he wants to leave, he can do so.
"We have had enquiries about him, one of which is from PSG.
"We'll take the time to discuss things with him. It's in his interest to find a club which will play him more and value him."
Lyon want a huge fee but Boro hope to drop the price and pay in stages as they seek the missing link in their side - with chairman Steve Gibson's cash.