Mr McConnell said the issue would instead be "front and centre" when the Senate reconvenes in September.
A long-time defender of gun rights, he shifted his tone by saying that failing to take action would be "unacceptable".
Democrats, however, urged President Trump to summon the Senate immediately to pass stricter background check laws.
Their party, which holds a majority in the House, passed measures on background checks in February. But it requires Republican support to get through the Senate and become law.
A letter backed by some 200 Democratic lawmakers urged the president to use his power to convene the Senate in extraordinary circumstances, saying that Mr McConnell "has been an obstacle to taking any action".
But speaking to a radio station in his home state of Kentucky, Mr McConnell said the president had called him on Thursday "anxious to get an outcome".
He said that calling the Senate back immediately would only result in "political point scoring" and he instead wanted to start discussions during the August break to come back with a coherent plan in September.
Democrats, however, continued to express concerns.
Representative Elijah Cummings, speaking to the National Press Club, said: "We really need to be careful when listening to politicians talk about what they're going to do," to some light laughter.
But striking a serious tone, he added: "You have a lot of talk… but in the end, nothing happens."
Mr McConnell said that background checks and so-called "red flag" laws - which allow a court to seize weapons from a person who may be a danger to themselves or others - were both on the table.
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"We've seen entirely too many of these outrageous acts by these mentally deranged people," he said, raising a Republican talking point that has been heavily criticised by political opponents.
In the wake of the two mass shootings just hours apart in El Paso, Texas, and Dayton, Ohio, Republican spokespeople began to blame mental health issues for the violence.
Asked about the availability of assault-style weapons as a factor, Mr McConnell said that would "probably be discussed" in September, but that background checks and "red flags" would be the focus.
Republican voices, including the president, also pointed to violent video games as a possible cause of the shootings, despite a lack of evidence that there is any such link.
Walmart, which had one of its stores attacked in the El Paso shooting, told its employees to remove any displays featuring shooting games, along with hunting videos and displays, Vice reported.
The company has not changed its policy on selling guns, despite a campaign urging them to do so.