BUDAPEST — The International Investment Bank, an obscure Russian financial institution with a small-time balance sheet, is an unlikely source of global intrigue. In more normal times, its plans to open a new headquarters in Budapest would pass unnoticed.

But the bank’s chairman has longstanding ties to Russian intelligence agencies. And the Hungarian Parliament has effectively granted the bank diplomatic immunity from any scrutiny by police or financial regulators — leaving Western security officials concerned that Russian spies could use it as a base to conduct European intelligence operations.

Several European nations have expelled Russian agents and cracked down on Kremlin finances in an effort to present a united front as Moscow is accused of meddling in the American election, poisoning a former spy in Britain and targeting European think tanks with hacking attacks.

And if the Europeans are now concerned about the Russian bank, so are the Americans, especially since the Trump administration has cultivated Hungary’s prime minister, Viktor Orban, as an ally. Days before Parliament granted immunity to the Russian bank, Secretary of State Mike Pompeo met Mr. Orban in Budapest and warned against allowing Russia to drive a wedge between friends.