Valverde, 57, is one of a handful of potential candidates to take charge until the end of the season before United hire a permanent manager.
United are focusing on an interim boss to replace Ole Gunnar Solskjaer.
They feel recruiting Paris St-Germain coach Mauricio Pochettino immediately may not be as straightforward as has been suggested.
Pochettino said on Tuesday he was “so happy” at PSG and that he will not be distracted by links to the United job.
Valverde fits the template of an experienced manager who may not expect to get the United job on a full-time basis.
The Spaniard’s last managerial role was with Barcelona, whom he guided to successive La Liga titles and the 2018 Copa del Rey before being sacked in January 2020.