Gundogan's shirt had "With respect for my president" written on it.
But he added the players "didn't help the DFB's integration work".
Erdogan is a keen soccer fan and as a young man played the game at a semi-professional level.
But rights campaigners say May is turning a blind eye to abuses, in Turkey's case to the jailing tens of thousands of people after a failed coup in July. Many German politicians have also criticised the footballers, questioning their loyalty to German democratic values.
"The United States, unfortunately, took its place without complaint alongside the Israeli government in this massacre of civilians and became a party to this crime against humanity", Prime Minister Binali Yildirim said.
A commentary in top-selling Bild, which ran the photos on its front page, said: "If you have a moral position, you don't laugh into the camera with Erdogan".
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The Joint Comprehensive Plan of Action (JCPOA) was signed by the US, China, Russia, Germany, France and Britain with Iran in 2015. In addition, Iran stated that in case of United States withdrawal from the nuclear agreement , they will also stop it.
With the Premier League season now all done and dusted, football fans are now switching their focus to this summer's World Cup in Russian Federation. The main question is whether the two German national players still fit into our German team.
"In that sense, his meeting with Germany internationals Mesut Ozil and Ilkay Gundogan as well as Turkey worldwide striker Cenk Tosun, who are all Germany-born professional footballers with Turkish descent, is only natural".
"I don't think we should involve them, they are Turkish but are playing for another national team".
Gundogan insisted he "didn't want to make a political statement with this photo and even less so campaign for his election".
Erdogan, in power for the last 15 years as premier and now president, will seek a new term in snap June 24 polls.
"We've always been clear that we want Turkey to uphold its global obligations, including respect for freedom of expression and political freedoms", he said. Erdogan also said he also would be discussing regional and worldwide issues with May.