An investigation is underway into the incident on board the vessel, which was based out of the Northern Fleet main port of Severomorsk in Murmansk.
Mr Putin told his defence minister to fly to Severomorsk and to thoroughly investigate the reasons for the fire, according to the Russian state-run news agency TASS.
Mr Putin said: "I ask you to fly to Severomorsk in order to personally listen to the reports and to aim the commission to get to the bottom of the causes of the tragedy."
He told Mr Shoigu to report to him personally on his return.
Mr Putin added: "I offer my sincerest condolences to the families of those killed. Everything possible must be done to help and support them."
The deep-sea research vessel was carrying out a military survey in Russian territorial waters.
The ministry's statement said the submersible is intended for studying the seabed, but didn't give its name or type. Authorities haven't given a cause for the fire, saying that an official investigation has started.
In the deadliest naval incident in post-Soviet Russia, the Kursk nuclear submarine exploded and sank on August 12, 2000, during naval manoeuvres in the Barents Sea, killing all 118 crew members.
Russian Defence Ministry reported. "On July 1, a fire broke out in Russian territorial waters on a research deep-water apparatus designed to study the bottom space and the bottom of the World Ocean in the interests of the Russian Navy.
"During the biometric measurements, 14 submariners died as a result of burning with poisoned products"
"The causes of the incident are established. The investigation is conducted by the Commander-in-Chief of the Navy."
In the US, Air Force 2, carrying Mike Pence, 60, was said to have been turned around and returned to Washington.
“There’s been an emergency callback,” Randy Gentry, a representative from Pence’s office, told the crowd.
“The vice president was asked to return to Washington so at this time we’re going to cancel today’s event.”
A White House spokesman said: “Something came up that requires the VP to remain in DC. There is no cause for concern.”
However Alyssa Farah, Mike Pence's spokeswoman, later denied there was ever an "emergency callback" and stated that Mr Pence never left Washington.
In a Twitter message she wrote: "The @VP never left Washington DC. There was no "emergency callback". Something came up that required the VP to stay in DC. We'll reschedule NH [New Hampshire] shortly."
However a White House official said there was no medical issues with either the vice president or Donald Trump.
Pence was due to take part in a roundtable discussion in Salem with former patients and others at the Granite Recovery Centre, an addiction treatment facility, on the drug crisis before flying back to Washington this evening.
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