President Obama -- NOT Buying Rancho Mirage House

There are lots of reports circulating that President Obama is in escrow to buy a house in the exclusive Rancho Mirage gated community which the likes of Frank Sinatra and Bob Hope called home -- NOT TRUE.

The house in question is listed at $4.25 million and escrow is supposed to close at the end of the month.

Sources close to the President who are familiar with the situation tell TMZ ... he and Michelle have never even seen the house and have definitely not made an offer. What's more ... we're told the Obamas will NOT live in the desert when his presidency ends.

And there's this ... the Obamas have relied heavily on their famed interior decorator Michael S. Smith. We're told 6 months ago someone who was looking at the house asked Smith to come along to give his opinion.