Jeroen Dijsselbloem, a 53-year-old former Dutch finance minister who also chaired the Eurogroup of euro zone finance ministers during the Greek financial crisis, is widely thought to be in the running to replace Christine Lagarde.
Bank of England governor Mark Carney and World Bank number two Kristalina Gorgieva are other names in the mix.
"We are backing one candidate who we believe has very good cards to become the EU candidate but I won't name names," Rutte told reporters at his weekly press conference.
Lagarde has been nominated to become the new European Central Bank president in October.
The IMF job is typically held by a European.
Dijsselbloem chaired the Eurogroup from 2013 until the beginning of 2018, leading dozens of lengthy emergency meetings during which bailouts for Greece, Cyprus and the Spanish banking sector were grudgingly pieced together.
The schoolteachers' son left national politics after his Labour party was heavily defeated in a 2017 election and is currently serving as chairman of the Netherlands' Safety Board, which investigates major accidents and disasters.