Both moves are dependent on Barca reducing their wage bill, though, with the Catalan club expected to be in excess of their LaLiga-imposed spending limit next season.
LaLiga president Javier Tebas has said they will need to reduce more than €150 million from their outgoings, with the Spanish league preventing them from registering midfielder Gavi's new contract until their financial situation improves.
Barca, meanwhile, are moving ahead with plans to strengthen and are targeting Athletic Club defender Martinez and Manchester City midfielder Gundogan, who are both out of contract in June.
Martinez, 31, has been tracked by Barca for years and the club have stepped up their interest in recent weeks due to the fact he is available for free at the end of the season.
Atletico Madrid are also keen on the Spain international, but sources have told ESPN that the centre-back has already given his word to Barca and that a transfer now depends on the Blaugrana cutting their wage bill.
A report from Jijantes claims Barca could include a clause which would see Martinez go out on loan, earning the salary agreed with Barca, if they are unable to register him.
Barca also want to reinforce in midfield, with Gundogan, 32, another player available for free, although sources have told ESPN that negotiations are not as advanced with the Germany international as they are with Martinez.
Talks have been ongoing for several weeks, though, and all parties are open to a deal in the right circumstances. Gundogan's agent, who also represents Barca defender Andreas Christensen, is a regular visitor to the club's training ground and was there in February for a meeting with director of football Mateu Alemany.
Barca, who are 12 points clear at the top of LaLiga, want to increase their squad depth ahead of next season after falling short in both the Champions League and Europa League this year.
Their financial situation has forced them to look for free agents, as they did with the signings of Christensen and Franck Kessie last summer, with another forward also wanted.
However, for players to arrive, others will have to leave. The effect of the assets sold last year, which funded the arrivals of Robert Lewandowski, Jules Kounde and Raphinha, will not be as strong next season.