McConnell's announcement follows the news that doctors have advised Senator John McCain to remain in Arizona for a week, following an operation Saturday to remove a blood clot above his left eye.
"Elaine and I, along with the entire Senate family, wish John the very best and wish him a speedy recovery," McConnell said in a statement. "While John is recovering, the Senate will continue our work on legislative items and nominations and will defer consideration of the Better Care Act."
McCain's absence in the closely divided Senate places the vote on the health care bill in jeopardy.
Senate Republicans need 50 votes to clear a procedural hurdle. They hold 52 seats in the chamber, but two senators - Rand Paul of Kentucky and Susan Collins of Maine - have already said they will not vote for the bill.
Last week, McConnell decided to cancel the first two weeks of the Senate's traditional August recess to allow more time for deliberations on the bill.
Senate Armed Services Committee Chairman Sen. John McCain speaks on Capitol Hill in Washington, May 23, 2017.
8203;McCain resting comfortably
McCain, who is 80, is "resting comfortably," according to a statement by an Arizona hospital.
The Mayo Clinic in Phoenix said in a statement that, following a routine physical, McCain had a procedure to remove a 5-centimeter blood clot located above his left eye.
"The senator is resting comfortably at home and is in good condition," the statement read. "Mayo Clinic doctors report that the surgery went very well' and he is in good spirits."
McCain's Senate office said in a statement that he "is in good spirits and recovering comfortably at home [in Arizona] with his family."
His doctors also have advised McCain to spend next week recovering from the procedure, the statement said.
McCain, a Vietnam veteran and former prisoner of war, was elected to the U.S. Senate in 1986. He was the Republican Party presidential nominee in 2008, losing to Barack Obama.