The move was supported by the Hillary Clinton campaign, which lost to the Republican presidential nominee Donald Trump, who is now the president-elect.
"The Electors require to know from the intelligence community whether there are ongoing investigations into ties between Donald Trump, his campaign or associates, and Russian government interference in the election, the scope of those investigations, how far those investigations may have reached, and who was involved in those investigations," the members said in the letter to the Director of National Intelligence.
The members comprise nine Democrats and one Republican.
Among the signatories include Christine Pelosi, daughter of the House Minority Leader Nancy Pelosi.
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In an unusual move, the Clinton Campaign has come out in support of the letter. "The bipartisan electors' letter raises very grave issues involving our national security. Electors have a solemn responsibility under the Constitution and we support their efforts to have their questions addressed," the Clinton Campaign Chairman John Podesta said in a statement.
"Each day that month, our campaign decried the interference of Russia in our campaign and its evident goal of hurting our campaign to aid Donald Trump. Despite our protestations, this matter did not receive the attention it deserved by the media in the campaign. We now know that the CIA has determined Russia's interference in our elections was for the purpose of electing Donald Trump. This should distress every American," Podesta said.