ISTANBUL—Turkey dismissed a U.S. call for an immediate cease-fire in northeast Syria, pressing ahead with its military offensive as senior Trump administration officials rushed to Ankara to try to halt fighting triggered by the withdrawal of American troops from the region.

President Trump on Tuesday dispatched Secretary of State Mike Pompeo and Vice President Mike Pence to the Turkish capital, where they will meet with President Recep Tayyip Erdogan on Thursday to urge him to stop a weeklong military incursion in northeastern...