By Noel Randewich

SAN FRANCISCO, Sept 20 (Reuters) - A move by the Trump administration to make it simpler to sell small arms abroad may provide some relief to gun makers American Outdoor Brands and Sturm Ruger & Company in an industry grappling with a deep sales slump since the election of President Donald Trump.

Shares of Sturm Ruger & Company and Smith & Wesson-owner American Outdoor Brands surged as much as 18 percent on Tuesday after Reuters reported a potential relaxing of oversight in foreign sales by the Trump administration, although they gave back over 4 percent in the following session.

The expected relaxing of rules could increase foreign gun sales by as much as 20 percent, the National Sports Shooting Foundation estimates. As well as the industry's big players, it may also help small gunsmiths and specialists who are currently required to pay an annual federal fee to export relatively minor amounts of products.

However, American Outdoor and Sturm Ruger, the two publicly traded U.S. gun makers, make well below 10 percent of their revenue from foreign sales, according to company filings, which means even a large increase in exports to other countries would have a minor effect on their overall businesses.

Still, analysts reacted positively to the plan, initiated but not completed by the Obama administration, to shift oversight of international non-military firearms sales from the State Department to the Commerce Department, which would put the focus more squarely on facilitating trade.

"While difficult to estimate the potential financial impact, we believe Smith & Wesson's iconic brand and products would sell well to international customers," Craig-Hallum analyst Steven Dyer wrote in a note to clients.

The companies did not immediately respond to a request for comment.

Fears that Democratic candidate Hilary Clinton would win the election and curb gun rights led to record purchases last year, but Trump's surprise victory has had a reverse effect, with 2017 sales slumping after consumers were no longer worried about new gun control measures.

Following Trump's election, gun stocks became a favorite of short sellers betting they would fall. Even after Tuesday's rally, American Outdoor Brands remains down 45 percent from before the election.

But suggesting that some traders believe gun stocks are near bottom, short interest in American Outdoor Brands has decreased by 43 percent since the end of June to $113 million, while short interest in Sturm Ruger has declined by 12 percent to $226 million, according to S3 Partners, a financial analytics firm.