Cornyn is the No. 2 Republican in the Senate behind Majority Leader Mitch McConnell, who said Tuesday morning that he'd recommended that President Donald Trump nominate Garland for the post.
Since firing Comey, Trump has raged at Democrats and other critics who have blasted him for firing the man who was spearheading an investigation centered around him.
A source familiar with Cornyn's thinking said the senator felt "obligated" to consider the job because a friend, Attorney General Jeff Sessions, asked him to, and Trump called and asked him to think about it.
Despite Cornyn's popularity - he's never lost an election, and is considered well-respected on both sides of the aisle - Suri believed he'd have difficulty being confirmed.
Sen. Chuck Schumer, the Senate minority leader from NY, told CNN's State of the Union that Comey's replacement should not be "a partisan politician - not in either party", and that it should be someone who is "experienced".
When asked whether Sessions involvement with the search raised any concerns - since Sessions had recused himself from the Russian Federation investigation - Cornyn said he did not see any conflict.
From a political landscape, Cornyn now serves as the Senate Majority Whip and Republicans control 52 Senate seats, both factors that may have lead some Republicans to try and convince Cornyn to hold his position.
Asked about the White House's shifting explanations for Comey's firing, Cornyn said he still backed the President's decision and supported the initial reasoning behind the firing - that Comey mishandled the Clinton email investigation.
Several Republican senators had promoted Garland even though they had refused to give him a hearing when Republican Trump's predecessor President Barack Obama, a Democrat, nominated Garland past year for a then-vacant seat on the US Supreme Court.
From a political landscape, Cornyn now serves as the Senate Majority Whip.
Cornyn has served as Texas' Attorney General, a Texas Supreme Court justice and a local judge.
One of the leading contenders to replace James Comey as the next director of the Federal Bureau of Investigation withdrew from consideration on Tuesday. Trump has said a decision could come before he leaves Friday for a trip to the Mideast and Europe. He is the second-ranking Republican in the Senate.
"I don't think that's a good strategy at all", Collins said, adding that she wanted to question Rosenstein Thursday on whether a special prosecutor was indeed needed.