* Suspect is widow of slain Islamist militant
* Security tightened nationwide before Olympics (Adds Putin-Obama phone call, U.S. offers assistance, paragraphs 4, 10-11)
MOSCOW, Jan 21 (Reuters) - Police have killed a senior Islamist militant in Russia's North Caucasus ahead of the Winter Olympics in Sochi, where security forces searched on Tuesday for a woman they fear is planning an attack during the games.
Eldar Magatov, a suspect in attacks on Russian targets and alleged leader of an insurgent group in the Babyurt district of Dagestan, died in a shootout, the National Anti-Terror Committee said on Tuesday.
President Vladimir Putin has staked his political reputation on organising a successful Olympics and tightened security nationwide after insurgents who hope to create an Islamist state in the North Caucasus threatened to attack the games, which start on Feb. 7.
Putin had a phone call on Tuesday with U.S. President Barack Obama in which they discussed how best to have a "safe and secure" Winter Olympics, the White House said.
In Sochi, which plans to host hundreds of thousands of visitors during the games, security forces were searching for Ruzanna Ibragimovaya, 23, who may have arrived to the Olympic host city on Jan. 11-12, a letter seen by Reuters said.
"(She) may be used as a terrorist-suicide bomber by (insurgency) leaders to organise terrorist acts during preparations for and during the 2014 Winter Olympics," read the letter, asking police to look immediately into the matter.
The letter from Russia's Federal Security Services to local police said she was the widow of a slain Islamist militant and is believed to have recently left her home in Dagestan, which lies nearly 600 km (370 miles) from Sochi on the opposite end of Russia's predominantly Muslim North Caucasus region.
Photographs of Ibragimovaya show a woman in a hijab, wearing a long dress.
Dagestan has become the focal point of the insurgency, which is rooted in two separatist wars in neighbouring Chechnya.
A White House statement about the Putin-Obama call said the two leaders talked about how best to advance shared U.S.-Russian interests, "including a safe and secure Sochi Olympics, for which the United States has offered its full assistance."
The U.S. military is making two ships in the Black Sea available should American support be required during the Olympics. Washington is also offering American technology used to thwart roadside bombs in places like Afghanistan to boost security during the Olympics, a spokesman for U.S Joint Chiefs of Staff Chairman Martin Dempsey said.
In a video published on Monday, two men said by an Islamist militant group to have carried out twin suicide bombings in the southern city of Volgograd last month warned Putin to expect more attacks if the games go ahead in Sochi.
The attacks in Volgograd killed at least 34 people and raised security fears before the Olympics. (Reporting by Thomas Grove; Additional reporting by Steve Holland, Jeff Mason and Phil Stewart in Washington; Editing by Timothy Heritage, Alister Doyle and Peter Cooney)