Quoting Russian official sources, Aljazeera reported that Moskoyev's brother and three other Chechen fighters were also killed.
Five Russian security personnel were also killed in the shootout on Wednesday in Argun, the province's third largest city, the sources said.
Aljazeera's reporter in Russia, Salam al-Ubaidi, said the operation was carried out by Russian intelligence officers after they located a group of Chechen fighters in the city.
In related news, the Russian Interior Ministry said two Chechen policemen and three members of the region's anti-terrorist force were killed in fighting with suspected Chechen fighters.
Three security officers were also wounded in the skirmish in Argun, east of the capital Grozny, ministry spokesman Roman Shchekotin said on Thursday.
The Associated Press put the number of suspected fighters killed at five. The Interior Ministry had initially reported two policemen dead in the fighting.
In other incidents, seven Russian soldiers were killed in gun battles with rebels in Chechnya, an official from the Moscow-backed government there said on Wednesday.
Four of the soldiers were killed and six others injured in the course of 19 separate attacks by Chechen rebels on Russian positions over a 24-hour period, said the official, who asked not to be named.
Another two soldiers were killed when the vehicle they were travelling in was blown up by a bomb near the southern village of Tsa-Vedeno.
Russian soldiers first took on the
province's rebels in Ocober 1999
One soldier was killed and two others were injured by a landmine explosion in Kurchaloi, southeast of the capital, Grozny.
In addition, two military engineers were injured trying to defuse an explosive device in Chechen-Aul, near Grozny.
Russian troops stormed Chechnya in October 1999 to try to re-establish control, following defeat in a first war against separatist guerrillas in 1994-96.