Republican leaders asked the Justice Department on Monday to open a criminal investigation into whether Hillary Clinton lied to Congress in testimony last fall about her private email server, opening a new front in their long-running attacks on the presumptive Democratic presidential nominee.

The Republican request, coming five days after the Justice Department closed a yearlong investigation into Mrs. Clinton’s handling of classified information in the emails, threatens to shadow her through the fall campaign and perhaps even into the White House if she is elected.

In a letter Monday evening, House Republicans asked the Justice Department to determine whether Mrs. Clinton had “committed perjury and made false statements” during her appearance in October before a special House panel on the 2012 attacks in Benghazi, Libya.