Sen. Kelly Ayotte said she is "disappointed" Lynch has refused to formally recuse herself.
Democrats say the measure doesn't go far enough.
Reid says it is "outrageous" lawmakers would leave without a deal.
"I think it didn't matter," Sen. Charles Schumer said. "I don't think she's lying."
Sen. Bob Corker (R-Tenn.) said "basketball has lost a legend."
McConnell tried to tee up a substitute amendment to a House bill to defund Planned Parenthood.
The Senate will take three votes on the bill.
Lydia Wheeler 06/29/16 02:45 PM EDT
The caucus claims there will be a July 12 hearing for the First Amendment Defense Act.
"Read our lips. ... We are not relenting," Sen. Debbie Stabenow said.
Sen. Mark Kirk said the administration must show it can prevent "terrorist infiltration."