“Football and the DFB (German football federation) defend values which are not fully taken into account by Mr Erdogan,” DFB president Reinhard Grindel tweeted.
“That's why it's not good that international players allow themselves to be manipulated for his electoral campaign.
“In doing that, our players have certainly not helped the DFB's work on integration.”
During a meeting Sunday with Erdogan, the two players, both born in the German city of Gelsenkirchen, handed over signed Arsenal and Manchester City shirts. Photos of the meeting were used on social media by Erdogan's campaign team.
Gundogan insisted he “didn't want to make a political statement with this photo and even less so campaign for his election.
“As German internationals we adhere to the values upheld by the DFB and we are aware of our responsibilities,” added Gundogan.
Earlier, the far-right German party, the AfD, reacted quickly to the picture, politician Beatrix von Storch asking: “Why is Gundogan playing for the German national team when he recognises Erdogan as his president?”
Ilkay Gundogan with Recep Tayyip Erdogan. Photo: DPA
Cem Ozdemir, former leader of the German Green party and himself of Turkish origin, was also scathing.
“The president of an international German footballer is called Frank-Walter Steinmeier, his chancellor Angela Merkel and his parliament the Bundestag, the headquarters of which are in Berlin and not Ankara,” Ozdemir told AFP subsidiary SID.
German national team manager Oliver Bierhoff said he would raise the issue with the players.
“I have no doubts over the desire of either Mesut or Ilkay to play for Germany and identify with our values,” said Bierhoff.
“Neither one of them was aware of the symbolic value of this photo, but it's clearly not right and we'll be talking to them about it,” he added.
Erdogan, in power for the last 15 years as premier and now president, will seek a new term in snap June 24th polls.
After the landmark vote, a new system agreed in an April 2017 referendum will come into force, which the opposition says will give the president authoritarian powers.
Turkish football federation vice-president Servet Yardmici, an executive member of the UEFA Board, claimed Erdogan's meeting with the players was centred on “football chat”.
Turkey, who are not qualified for the 2018 World Cup, is one of several countries hoping to host the 2024 European Championships, but Yardmici — who described the players in the photo as “Turkish” – affirmed none of them would feature in any upcoming promotional campaign.
“I don't think we should involve them, they are Turkish but are playing for another national team. We should not involve them in our campaign,” he told AFP in London on Monday.
Asked about the photo, he added: “Our president Erdogan is visiting England and was an active footballer and he loves meeting Turkish football players especially, although he does meet foreign players in Turkey when he has the time.
“That is one of the reasons he met them yesterday (Sunday), to ask them how many goals they scored etc, it was football chatting but I have to make it clear they are not part of our campaign and there is no intention of bringing them into our campaign.”