WASHINGTON—House Democrats earlier this week released records asserting that Rudy Giuliani was in contact with a phone number associated with the White House budget office, where $400 million in aid to Ukraine was temporarily put on hold this summer.

That assertion, however, is now in doubt. The previously undisclosed phone number cited by the House Intelligence Committee in its impeachment inquiry report, which The Wall Street Journal has identified, isn’t directly linked to the Office of Management and Budget. It could instead...