"I asked them to reflect on the subjects the president raised within their jurisdictions and encouraged them to engage in bipartisan discussions of potential solutions to help protect our communities without infringing on Americans’ constitutional rights," McConnell said.

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McConnell's comments came after Trump called on the country to condemn white supremacy following the back-to-back mass shootings and threw his support behind new measures focused on mental illness rather than stricter gun laws.

Trump also talked up the need for lawmakers to work in a bipartisan fashion to respond to the shootings, saying, "We must seek real, bipartisan solutions."

"We have to do that in a bipartisan manner. ... Republicans and Democrats have proven that we can join together in a bipartisan fashion to address this plague," Trump said.

McConnell added in his statement that "Senate Republicans are prepared to do our part."

Wicker, in a statement, confirmed that he had spoken with McConnell and echoed the GOP leader's call for bipartisanship, saying, "It will be important for any solution we consider to be able to pass the Senate and the House and earn the president’s signature."

Alexander, in a string of tweets, said he was "ready to do more, especially on background checks, to identify those who shouldn’t have guns."

Some GOP senators have talked up the need for legislation in the wake of the shootings in El Paso, Texas, and Dayton, Ohio, which left at least 31 people dead.

McConnell is also under growing pressure from Democrats to call the Senate back from the August recess in order to work on gun legislation, which faces an uphill battle in the GOP-controlled chamber.

McConnell took a veiled shot at the Democratic rhetoric on Monday without directly responding to calls to reconvene the Senate.

"Only serious, bipartisan, bicameral efforts will enable us to continue this important work and produce further legislation that can pass the Senate, pass the House, and earn the president’s signature," he said.

"Partisan theatrics and campaign-trail rhetoric will only take us farther away from the progress all Americans deserve," he added.