The Pentagon put Turkey on notice that it will cut off Turkish military pilots from training and halt any Turkish purchases of the F-35 fighter jet, defense officials said Friday, exerting new pressure in a widening rift between the two allies.

Acting Defense Secretary Pat Shanahan told his Turkish counterpart in a memo released Friday that if Ankara makes good on plans to purchase a Russian antiaircraft system, the U.S. would take steps to discontinue Turkey’s participation in the development and use of the plane.

The...