ISTANBUL—President Recep Tayyip Erdogan on Friday warned he wouldn’t tolerate higher lending rates for very long, challenging the country’s central bank and raising the stakes on an economic plan meant to calm the country’s currency crisis.

Finance Minister Berat Albayrak—Mr. Erdogan’s son-in-law—is due to present the plan next Thursday to explain how Turkey will deal with the tremors caused by a 38% drop in the Turkish lira against the dollar this year.

The...