Wild card Murray was 7-5, 4-1 down to Yannick Hanfmann when a thunderstorm lashed the Parc des Eaux-Vives grounds and the match was called off for the night.
Ever since Friday's draw, tennis fans had been relishing what could be the final match between members of the Big Four who have dominated men's tennis this century: Djokovic, Murray, Rafael Nadal and the now-retired Roger Federer.
Murray, 37 and playing with a metal hip, has said he is unlikely to carry on playing competitive tennis beyond the next few months, while Nadal, also plagued by injuries, is likewise facing the end of his career and is unsure if he will make a farewell appearance at Roland Garros next week.
Now the world number 75, Murray was facing the German world number 85 in the Geneva first round -- with world number one Djokovic awaiting the winner in the second.
The 28-man clay court tournament serves as a final tune-up before next week's French Open, the second Grand Slam of the year.
Former world number one and three-time Grand Slam winner Murray was playing his first tour-level match since damaging tendons in his left ankle in Miami in March.
His match with Hanfmann will resume Tuesday and the winner will face Djokovic on Wednesday.
Djokovic took a wild card to play in Geneva in a bid to rescue an alarming dip in form ahead of his French Open title defence next week.
The record 24-time Grand Slam champion will arrive in Paris without a title in the season for the first time since 2018, unless he takes the Geneva trophy.
Casper Ruud, the Norwegian world number seven, is also in the Geneva draw, along with fellow top 20 players Taylor Fritz, Ben Shelton and Sebastian Baez.
Murray and Djokovic last played in 2017 in the Doha final, with the Serb taking the title.
Murray won the US Open in 2012 and Wimbledon in 2013 and 2016, making 11 Grand Slam final appearances in all.
