Priebus said that the RNC would pay for five security guards to keep the memorial open to veterans and the public.
Priebus accused the White House of closing the memorial to make the shutdown painful, and suggested the administration would say it isn’t legal to accept the RNC offer.
“The Obama administration has decided they want to make the government shutdown as painful as possible, even taking the unnecessary step of keeping the Greatest Generation away from a monument built in their honor,” Priebus said.
“That’s not right, and it’s not fair," he said. "So the RNC has put aside enough money to hire five security personnel to keep this memorial open to veterans and visitors. Ideally, I’d hope to hire furloughed employees for this job."
Priebus also called on the Democratic National Committee (DNC) to join them in helping pay for the memorial to remain open.
The DNC responded to Priebus's offer, calling it a "silly stunt."
DNC Communications Director Mo Elleithee said in a statement that Republicans should pass a clean stopgap funding measure if they want to reopen the monument.
"It sounds like the votes are there if the Speaker would just call for a vote," Elleithee said. "It would save the economy a lot of money and get the memorial and government open a whole lot faster."
The World War II Memorial has become a political flashpoint during the shutdown, as Republican lawmakers helped touring veterans’ groups to enter the memorial on Tuesday.
Republicans have slammed Democrats and President Obama for voting against a piecemeal funding bill that would fund the National Park Service.
White House press secretary Jay Carney on Wednesday accused Republicans of grandstanding over the issue.
The White House spokesman said Republican lawmakers who acted "as if they were surprised" that the monument would shut down "clearly didn't pay attention."
"The fact is, when you shut down the government, you shut down a lot of services," Carney said.
On Wednesday, the National Park Service said it would allow the memorial to remain open to veterans because their visit was a First Amendment activity. The memorial remains closed to the public, however.
World War II-era veterans are touring the memorial through Honor Flight, an organization that brings veterans to Washington from across the country.
The World War II Memorial and other open-air monuments on the Mall are normally open 24 hours a day, but fences have been placed around them after the government shutdown.
--Justin Sink contributed to this report.
--This report was updated at 3:30 p.m.