The 20-year-old Brazilian forward has set off a race between two Championship clubs who want to sign him from Flamengo.
Middlesbrough are serious about taking the forward to Teesside in the ongoing transfer window and have held talks for his signature at a luxury hotel in Lisbon.
Boro met emissaries from Flamengo in Portugal on Monday to discuss the possibility of taking him to England.
Fulham have also been keeping tabs on Muniz and, according to Brazilian outlet UOL, the Cottagers have gone in with a concrete bid to sign the striker.
The London club have offered to sign him on loan and are ready to include a mandatory purchase clause in the agreement.
New Fulham boss Marco Silva has expressed his specific interest in signing Muniz from Flamengo this summer.
Middlesbrough also remain keen and are hoping to beat Fulham to the forward’s signature.