The leaders spoke on the phone and Mr Sunak “reiterated the importance of the UK-Turkey relationship, with close and growing ties in trade, investment, and defence”, a Downing Street representative said.
Mr Erdogan updated Mr Sunak on the planned visit by Ukraine's President Volodymyr Zelenskyy to Turkey.
Mr Sunak acknowledged Turkey's role in mediating between Russia and Ukraine, and its continuing support for the Black Sea Grain Initiative amid war between the two countries.
As the Nato Summit approaches next week, Mr Sunak said he was looking forward to collaborating with Mr Erdogan on shared challenges, to ensure the alliance remains robust and agile in protecting collective security.
The British Prime Minister voiced his desire for the alliance to proceed with the ratification of Sweden's Nato accession soon.
He said there were significant benefits in Sweden's membership and progress had been made in addressing Turkey's legitimate security concerns.
“The Prime Minister also highlighted opportunities for greater collaboration on tackling illegal migration, an issue which affects both our countries,” the Downing Street representative added.
“The UK would welcome efforts to further strengthen our co-operation on tackling criminal people smuggling gangs and returns.”
Mr Sunak had called Mr Erdogan to congratulate him on his re-election, reiterating the “strong relationship” between the UK and Turkey, recognised as economic partners and close Nato allies.
The British Prime Minister offered continued British solidarity amid Turkey's continuing recovery from the devastating earthquakes experienced earlier this year.
Downing Street said the leaders agreed to continue working closely together to address shared challenges.