The 27-year-old Scot will now take on the world No 41 at 9am (BST) and would then have to play a semi-final later in the day if he wins.
The world No 3 had kicked off his clay-court season with a trouble-free straight sets win over Germany’s Mischa Zverev on Thursday.
Murray, who has been playing under new coach Jonas Bjorkman for the first time, is bidding to land his first ATP clay-court title of his career.
Rosol, who beat Florian Mayer and Sergiy Stakhovsky to reach the last eight, trails Murray 0-1 in their head-to-head, although he did take the first set off him in their only encounter last year in Indian Wells.
Roberto Bautista Agut’s last eight clash with Victor Estrella Burgos has also had to be postponed 24 hours.
Organisers are still hoping the rain will relent to allow the quarter-finals from the second half of the draw – Dominic Thiem v Gerald Melze and Philipp Kohlschreiber v David Goffin – to go ahead as planned.