Flake told CNN on Tuesday that he voted against a Senate amendment on background checks negotiated by Sens. Joe ManchinJoseph (Joe) ManchinDems mull big changes after Brazile bombshell A bipartisan bridge opens between the House and Senate Collins, Manchin to serve as No Labels co-chairs MORE (D-W.Va.) and Pat Toomey (R-Pa.) because it would have made background checks on gun sales over the Internet too costly.

He said he was willing to change his position if his concerns about Internet sales were met. 

After the CNN report, Flake posted a message on Twitter that underlined his continued opposition to Manchin-Toomey:

Flake's comments come as Democrats seek to hold another vote on the gun legislation. In April, the so-called Manchin-Toomey bill failed to win the 60 votes it needed to pass the Senate. Flake was one of the 46 senators that voted against the bill.

Since voting against the measure, Flake has faced criticism from groups advocating stricter gun laws. A recent poll also found Flake's approval rating at 32 percent. Flake suggested his vote on the background check legislation was largely to blame for the low approval numbers.