Robertson, 32, is leaving Liverpool as a free agent when his contract expires at the end of the season, bringing to an end a nine-year stint at Anfield during which he became one of the most decorated left-backs in the club's history.
Fenerbahçe had invited his agent to Istanbul to present a verbal offer, and the Scotland international is understood to have given the proposal consideration before talks stalled.
President Sadettin Saran's decision to call elections effectively suspended the club's ability to commit to transfer agreements, with no contracts or deals expected to be finalised until the new leadership structure is in place.
Tottenham Hotspur came close to signing Robertson in the January transfer window, with the two clubs agreeing a fee in the region of £5 million before Liverpool decided not to sanction a mid-season departure.
Tottenham remain interested, subject to retaining their Premier League status. Robertson is also expected to attract interest from other Premier League clubs and from Italy.
Robertson made his final Anfield appearance against Crystal Palace last weekend, scoring the second goal in a 3-1 win before Mohamed Salah was forced off injured.