Cruz called choice "a first principle" in any education reform effort.
Advocates told Sen. Mitch McConnell he is "standing in the way of progress."
Senators are expected to vote around 5:30 p.m. on a judicial nominee.
The Foreign Relations Committee passed Iran legislation this week.
Sen. Jeanne Shaheen (D-N.H.) is looking for recommendations on who should get the spot.
Jordain Carney 04/17/15 06:01 AM EDT
His maneuvers on Lynch shed light on how he will lead the upper chamber in this Congress.
Senators also approved Jonodev Osceola Chaudhuri as National Indian Gaming Commission chairman.
A Republican aide said senators are "closer to a deal than we have been in the past."
“We're involved right now in active negotiations,” said Sen. Debbie Stabenow (D-Mich.).
"Republicans haven't had a desire to govern dependably," the Democrat said.