The Turkish parliament's foreign affairs commission approved the bid last week in a key step towards enlarging the Western bloc after 19 months of delays. The next step is a vote in the parliament's general assembly.
Blinken raised the Swedish bid on the first day of a week-long regional tour to address the Gaza conflict.
"In Turkey, we also talked about the final steps in the process to ratify Sweden's accession to NATO in the coming weeks," he told reporters in Greece, but gave no details.
Sweden made the bid in 2022 after Russia's invasion of Ukraine but Turkey's President Tayyip Erdogan raised objections over what he said was Stockholm's protection of people who Ankara deems to be terrorists.
U.S. lawmakers have held up the sale of F-16 fighter jets to Turkey until it signs off on Swedish membership.
(Reporting by Simon Lewis and Tuvan Gumrukcu; Writing by Daren Butler and David Ljunggren; Editing by Jan Harvey and Chris Reese)
