WASHINGTON — The Senate Intelligence Committee has concluded that Republicans on the House Intelligence Committee were behind the leak of private text messages between the Senate panel’s top Democrat and a Russian-connected lawyer, according to two congressional officials briefed on the matter.

Senator Richard M. Burr of North Carolina, the committee’s Republican chairman, and Senator Mark Warner of Virginia, the top Democrat, were so perturbed by the leak that they demanded a rare meeting with Speaker Paul D. Ryan last month to inform him of their findings. They used the meeting with Mr. Ryan to raise broader concerns about the direction of the House Intelligence Committee under its chairman, Representative Devin Nunes of California, the officials said.

To the senators, who are overseeing what is effectively the last bipartisan investigation on Capitol Hill into Russia’s interference in the 2016 election, the leak was a serious breach of protocol and a partisan attack by one intelligence committee against the other.

The messages between Mr. Warner and Adam Waldman, a Washington lawyer, show that the senator tried for weeks to arrange a meeting with Christopher Steele, the former British spy who assembled a dossier of salacious claims about connections between Mr. Trump, his associates and Russia. The Senate committee has had difficulty making contact with Mr. Steele, whom it views as a key witness. And Mr. Waldman, who knew Mr. Steele, presented himself as a willing partner.