Nearly 100 people, including civilians, have died in three days of fighting over the disputed mountainous region.
The enclave is internationally recognised as part of Azerbaijan, but has been run by ethnic Armenians since a 1988-94 war between the two former Soviet republics.
Azerbaijan has repeatedly stated that its air force does not have F-16 fighter jets. However, Turkey does.
The fighting that started three days ago now appears to be spilling out of Nagorno-Karabakh, with Armenia and Azerbaijan trading accusations of direct fire into their territories.
They also blame each other for starting the conflict.
While Turkey openly backs Azerbaijan, Russia – which has a military base in Armenia but is also friendly with Azerbaijan – has called for an immediate ceasefire.
What’s happened to the jet?
Armenian Defence Ministry spokeswoman Shushan Stepanyan said the Armenian SU-25 was shot down on Tuesday morning and the pilot “died heroically”.
In a Facebook post, she said the Turkish F-16 was 60km (37 miles) deep into Armenian air space.
Turkey immediately denied the claim as “absolutely untrue”.
“Armenia should withdraw from the territories under its occupation instead of resorting to cheap propaganda tricks,” Turkish President Recep Tayyip Erdogan’s aide Fahrettin Altun said.
No material evidence about the jet has yet been released. Azerbaijan has called for Armenia to provide it.
What is the latest from the battlefield?
Earlier on Tuesday, both Armenia and Azerbaijan said that heavy fighting had continued overnight in Nagorno-Karabakh.
The region’s self-proclaimed authorities said 87 of their military personnel had been killed and 120 wounded since the fighting began on Sunday, according to the Armenpress news agency.
They put the number of deaths on the Azerbaijani side at nearly 400, saying that one aircraft, four helicopters and a number of tanks had been destroyed.
Azerbaijan has released no figures on its military casualties, but says 12 civilians were killed by Armenia fire.
Azerbaijan’s defence ministry was quoted by news agencies as saying that Armenian troops had repeatedly tried and failed to regain lost positions in the Fuzuli-Jabrayil and Aghdere-Terter areas.
The ministry said that a column of Armenian armoured and other combat vehicles had been destroyed, adding that the enemy had suffered heavy losses.
The casualty claims by Armenia and Azerbaijan have not been independently verified.
What chance of a ceasefire?
There appears to be little at the moment.
Armenian Prime Minister Nikol Pashinyan told Russian media the atmosphere was not right for talks while military operations were ongoing.
Azerbaijan’s President Ilham Aliyev, also speaking to Russian outlets, ruled out any talks given Armenia’s current stance.