Top Republican senators are tightening the screws on Secretary of Veterans Affairs Eric Shinseki to resign after allegations that up to 40 people died due to delayed care at a VA hospital in Phoenix, Arizona.

"General Shinseki's time as Secretary of the Veterans Affairs has come to an end and he needs to step down," Sen. John Cornyn, the Senate's second-highest ranking Republican, said at a news conference today. "The president needs to find a new leader to lead this organization out of the wilderness and back to providing the service that our veterans deserve."

"I ask the Secretary to submit his resignation, and I ask President Obama to accept that resignation," Sen. Jerry Moran, R-Kansas, said on the Senate floor.

Senate Minority Leader Mitch McConnell did not explicitly call for Shinseki's resignation, but did say "a change in leadership might be a good step in the right direction" after a "stunning period of dysfunction" at the VA.

The American Legion, the country's largest veterans group, called for Shinseki to resign on Monday.

Senate Majority Leader Harry Reid defended Shinseki today, calling him a "fine man, a dedicated patriot to our country."

"The issue that came up in Phoenix, these are allegations and there will be a complete investigation of that, what's gone on - whether there's substance to it or not, but it certainly doesn't call for the general to resign," Reid said.

Shinseki, a former Army general who earned two purple hearts while serving in Vietnam, has lead the Department of Veterans Affairs since 2009.