The Russian Football Union (RFU) stressed Fabio Capello remains under contract with the national team.
Shakhtar Donetsk claim there has been no approach from Liverpool for in-demand midfielder Henrikh Mkhitaryan.
The 31-year-old has established himself as a firm fans' favourite at St Mary's since joining from Bristol Rovers in August 2009.
Lambert has scored 101 goals for Southampton and his 13-goal haul this season makes him the top English scorer in the Barclays Premier League.
The striker's contract had been due to expire at the end of next season but has now been extended by a further two years.
"I'm over the moon to sign with Southampton for another three years," Lambert told the club's official website, www.saintsfc.co.uk.
"I am already enjoying myself massively at this club and the aim now is to stay on for another three years and help push the club on further than where it already is.
"There is a lot of ambition in this club for the next few years and that is one of the main reasons that I have signed the contract - and that is why I want to end my career here.
"The core of players who have been together for two or three years are going to be here for another three or four years and that is a really good feeling.
"We all have the same ambition and [executive chairman] Nicola Cortese's ambition for this club drives us all on as well, so it really helps the club with where we are now and it will help with where we want to go."
Lambert's contract extension follows recent new deals handed to Morgan Schneiderlin, Jason Puncheon and Kelvin Davis - others to have played a key part in Southampton's rise from npower League One to the top flight.
"Rickie Lambert is a very special player for Southampton Football Club," Saints manager Mauricio Pochettino said.
"To have scored over 100 goals is a remarkable achievement, and one which he has made possible thanks to hard work and dedication.
"In my time at the club I have been impressed by his mentality and commitment to improving himself in training - attributes that have allowed him to adapt to the Premier League so well.
"I am delighted that Rickie has committed himself to the club for a further three years, and have no doubts that he will continue to deliver for Southampton as we look to push on to the next level."