By Jacob Gronholt-Pedersen and Stine Jacobsen

COPENHAGEN (Reuters) - Denmark's government has proposed purchasing two new Arctic inspection vessels and increasing dog sled patrols to boost its military presence in Greenland, as U.S. President-elect Donald Trump sets his sights on the island, local media reported.

The government has also proposed upgrading the airport in Kangerlussuaq, a former U.S. military base in western Greenland, to accommodate F-35 fighter jets, broadcasters DR and TV2 said on Friday.

Denmark has already set aside $400 million to strengthen surveillance and intelligence in the Arctic and North Atlantic with long-range drones.

Trump this week reiterated his ambition to gain control over Greenland, a semi-autonomous Arctic territory of Denmark that the incoming U.S. president views as crucial to American security.

Denmark, while responsible for Greenland's security and defence, has limited military capabilities on the vast island. At present, these include four ageing inspection vessels, a Challenger surveillance plane and 12 dog sled patrols, all tasked with monitoring an area four times the size of France.

The tradition of using sled dogs for patrolling Greenland dates back to World War II, with the patrol being a specialized unit of the Royal Danish Navy.

DEFENCE SPENDING TALKS

Following more than a decade of drastic cuts in defence spending, last year Denmark allocated 190 billion Danish crowns ($26 billion) for its military over a ten-year span, some of which will be for the Arctic.

Danish lawmakers began long-delayed talks on Friday on how to apportion those funds, which are being progressively allocated via political agreements. The discussions that commenced on Friday focused on what portion of the budget should be go to Greenland's defence requirements.

Earlier this week, Denmark's Foreign Minister Lars Lokke Rasmussen said he looked forward to informing Trump about Denmark's "massive" military build-up since his last term as president.

However, Defence Minister Troels Lund Poulsen acknowledged on Thursday that necessary military investments had been "neglected for many years."

Poulsen last month said more funds should be allocated towards Denmark's military presence in the Arctic, although he did not provide specifics.

"The U.S. concerns are real, and there are some very concrete, huge problems in relation to Greenland," defence analyst and retired navy commander Jens Wenzel Kristoffersen told Reuters.

"The U.S. needs to feel safe in this area, and if Denmark doesn't do something about this, then they will have to deal with it themselves."

The U.S. military maintains a permanent presence at the Pituffik Space Base in Greenland's northwest. The strategic location is vital for the U.S. military and its ballistic missile early-warning system, as the shortest route from Europe to North America runs via the island.

On Thursday, the U.S. embassy in Copenhagen stated that there were no plans to increase the U.S. military presence in Greenland.

($1 = 7.2425 Danish crowns)

(Reporting by Jacob Gronholt-Pedersen; Editing by Sharon Singleton)