Lodi spent the season on loan at the City Ground, while Navas was brought in on deadline day in January to cover for another loanee Dean Henderson, who suffered a season-ending injury the same month.
Forest are also on high alert about potentially bringing Henderson back to the club, with Manchester United closing in on the signing of Andre Onana.
A goalkeeper will be on the wishlist for Steve Cooper this summer, with Wayne Hennessey and Ethan Horvath the only two keepers currently at the club.
Horvath spent last season on loan at Luton Town and his future remains up in the air, while Hennessey is not expected to start the season as Forest's number one.
Henderson was a regular until his injury in January, before Navas came in to deputise and helped the side towards survival in the Premier League.
However, there are hopes that the club could welcome the three-time Champions League winner back to the club, as claims from L'Equipe state that Paris Saint-Germain have made Navas available for a permanent transfer, with Gianluigi Donnarumma unlikely to be dislodged as number one.
There would be a small transfer fee involved, but Navas's wages could be a potential stumbling block in the deal though.
That may open the door for Fenerbahce to make a move for the Costa Rican, with FOX Deportes reporting that the player could be tempted by a move to Turkey.
A similar fate is set to strike the club in terms of a move for Lodi, as Forest are likely to be priced out of a move for the Atletico Madrid left-back.
Lodi remains a sporadic member of the Brazilian national side, and recent reports from Andres Onrubia Ramos of Diario AS claims that Atletico are looking for around £24m for the player.
After making 28 appearances at the City Ground last season, and with Reinildo ahead of him in the pecking order in Madrid, a return to Forest appeared likely for Lodi.
However, FootMercato claims that Marseille have moved ahead and now agreed terms on a deal that will see the 25-year-old make a move to France, with Lodi set to come in as the replacement for outgoing pair Sead Kolasinac and Nuno Tavares.
One deal which could still realistically happen though is for Henderson, and Forest's chances of welcoming the Man United man back to the club were given a boost recently.
The Daily Mail claims that Henderson has told the club he is not prepared to be second choice for the upcoming season, so if a move for Onana is completed, then he will expect to depart.