Think "Keeping Up with the Kardashians" meets "Raiders of the Lost Ark" -- in a new lawsuit, the government claims a statue dating back to 1st or 2nd Century Rome ended up getting shipped to the U.S. in 2016.

The problem for Kim is the paperwork listed "Kim Kardashian dba Noel Roberts Trust" as the consignee and importer of said statue. Noel Roberts is, in fact, a trust Kim has used to make other big-ticket purchases -- but a rep for Kim denies she made this deal.

A source connected to the situation tells us Kim only found out this week that the piece was actually bought by Kanye as part of a larger purchase which included more art. We're told it's unclear if Kanye had bought it as a gift, or when he had planned on telling Kim, but our source says she was in the dark.

Now, the reason the feds want it back is because Italy's determined the piece is a cultural treasure that was essentially looted from an archaeological site.

As for the alleged Kardashian ties -- well, there's one smoking gun. The feds say a Belgium designer named Axel Vervoordt shipped the statue to the U.S., and presumably, his company put Kim's name on the customs document. Vervoordt has worked with Kim and Kanye in the past, and back in 2016 -- when the statue was imported -- they were building their Hidden Hills mansion ... which has been compared to a museum.

Still, Kim says she has no knowledge of the statue or its purchase. For his part, Vervoordt tells TMZ ... he acquired the statue "in good faith" from a French gallery, which in turn says it bought it from a German auction house. He adds, "At this point there is no evidence that this piece was illegally imported from Italy. If investigation proves that the piece was actually exported from Italy without a proper export license from the Italian Ministry of Culture, then we will of course take our responsibility."

A source close to Kim says that while she doesn't know the origins of the work, she hopes it gets resolved ASAP.