The Dubai club had a bid of £850,000 rejected by West Brom for Odemwingie last month, with the player currently in limbo at The Hawthorns
Odemwingie is out of favour under Steve Clarke after famously failing to engineer a transfer deadline day move to Queens Park Rangers in January despite driving to Loftus Road.
A number of Premier League clubs are thought to have expressed an interest in Odemwingie, but Al Nasr are hoping to win the race for the Nigerian international.
Al Nasr are thought to have increased their initial offer up to £1.5million and the possible deal could speed up this week with the chairmen of both clubs expected to hold talks in an attempt to find a resolution.
West Brom are happy to sell Odemwingie, but insist no deal will be done until their valuation is met for the 32-year-old.