Rep. Tom Cole (R-Okla.) said Republicans can't vote to sue President Obama for overstepping executive authority and then fail to pass a bill allowing him to handle the border crisis. 

"You can't say on the one hand the president is overreaching by acting without legislative authority and direction and then refuse to give him legislative authority and direction in other areas," Cole said on MSNBC's "Morning Joe."

Cole, a deputy whip, said he was confident a border bill would eventually pass the House after Republicans were forced to pull their $659 million proposal Thursday for a lack of support, though he said passage might not come before the August recess.  

Cole was commenting on a statement House leadership put out shortly after pulling its own bill dealing with the surge of child immigrants being detained at the Texas-Mexico border. 

Democrats pounced on one part of the GOP statement, accusing the party of being hypocritical for calling on Obama to take a number of steps without congressional action. House Republicans earlier in the week voted to sue Obama for overstepping his authority by delaying parts of the Affordable Care Act.

"Well I'm not going to disagree with you, because it is a point I made myself in conference," Cole said about the criticism.

Some of the GOP recommendations called only for Obama to send a strong public message to deterring illegal immigration or roll back a number of programs Republicans believe are contributing to the influx. 

Cole expressed confidence that the House would pass a bill, saying "the speaker made it very apparent." The timeline could be anywhere between Friday and Sept. 5, he said.  

He criticized Democrats in the Senate for leaving town after its border proposal failed to advance. 

The Senate is in session Friday, but no votes are scheduled.

"Harry ReidHarry Mason ReidLobbying world Senators fume over fight to change rules for Trump's nominees After Dems stood against Pompeo, Senate’s confirmation process needs a revamp MORE and Democrats in the Senate couldn't pass a bill yesterday and they quit and went home," he said. "We couldn't pass a bill yesterday and we're at work and we are here."