A Downing Street spokesman said a "frank discussion" on human rights was on the agenda for the meeting.
Aktay also stressed that Erdogan, who is a former football player, met the players in a personal capacity and the meeting had nothing to do with the June elections.
Minor scuffles broke out as pro-Erdogan counter-protesters walked in front of Kurdish demonstrators outside Downing Street, with police wrestling some of the ringleaders to the ground.
Last month he declared snap elections for June 24, bringing the polls forward by more than a year.
The move is widely regarded as a plan to shore up his power in the country and comes as worldwide pressure mounts on the president over his suppression of political dissent.
Among the protesters in London on Tuesday were journalism groups calling for the release of detained media workers.
He criticised Germany by making an apparent reference to dissident Turkish journalist Can Dundar's meeting with German President Joachim Gauck late previous year.
"Just because they are journalists, just because they have a press card, doesn't mean everything is allowed", he said.
"We must make it clear to our government here that we do care about the fundamental freedoms - these values that our own country is anchored upon", Reporters Without Borders UK chief Rebecca Vincent told protestors.
"Under the state of emergency people do not have full access to independent information, so you now have an electorate that will be going to the polls again after quite a short campaign period without having the full picture".
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Speaking on Tuesday, Erdogan welcomed a transition deal agreed between London and the European Union to ease Britain's departure from the bloc.
It's only the latest spat between Germany and Turkey since a failed coup attempt against Erdogan in 2016.
They were chanting slogans and holding posters that described Erdogan as a "terrorist" for his government's actions at home and in Syrian Kurdistan.
In a speech at a charity event in London on Sunday, Erdogan painted a different picture, saying that his country is struggling to protect "all the oppressed and otherised" whose "future has been taken away".
Gundogan says he didn't intend to make a political statement, and Germany's soccer officials and leaders seem content to accept his gesture to Erdogan as an innocent mistake.
His government arrested more than 50,000 people following a July 2016 coup attempt, and some 38,000 people still are behind bars for alleged links to the failed overthrow.
"He is the publicly elected president of Turkey".
The Turkish head of state also touched upon the transfer of the U.S. Embassy to Jerusalem.
"All these murders committed by Israel and, unfortunately, the soil that created US" - said President Erdogan.
"We will continue to stand with Palestinian people with determination", he added.