April 7 (Reuters) - Chicago Federal Reserve Bank President Austan Goolsbee said on Tuesday he is worried that the Iran war will drive inflation higher even as it slows the U.S. economy, putting the Fed in an uncomfortable position where there is no obvious "cookbook" for what to do. 

"The prices spiked from tariffs and they were supposed to go away, and this is now hitting before that went away," Goolsbee said at the Detroit Economic Club, referring to the surge in oil prices since the war began on February 28.

The longer oil prices stay high, the more deeply inflation may get embedded into the economy, Goolsbee said, pushing up prices but also threatening a job market that for now is "stable but not great."

"You're just in a very uncomfortable situation and there's not an obvious cookbook of should we ... heat things up or cool things down. It's not obvious which way to do it," Goolsbee said.

The Fed left short-term interest rates on hold in the 3.5%-3.75% range last month, and had signaled it could deliver another interest-rate cut later in the year if inflation resumes progress toward the central bank's 2% goal. Policymakers next meet later this month.  

"The possibility of a stagflationary outbreak coming from high oil prices before the tariff inflation went away, leading to the main engine of growth - the U.S. consumer - just giving up and saying we don't have confidence, we're going to start hoarding our money, and sending us into a stagflationary recession - that'd be the worst outcome," Goolsbee said on Tuesday.

Of the national economy, Goolsbee said, "I'm cautious - slash nervous - about it in the moment."

In an interview later on WJR 760AM radio, Goolsbee said that given that lack of clarity, the debate at the Fed's policysetting table can be contentious. 

Financial markets are betting the Fed will leave rates on hold through the end of the year.