Cohen has previously been linked with a move to Leicester - but Northampton confirmed tonight they will not be terminating his contract, which still has one season left to run.
"Ben's contract with the club runs until the end of the coming season and he is not able to play for any other club in the meantime," said Northampton director of rugby Jim Mallinder.
Cohen, who made his Saints debut in 1996 and won the Heineken Cup in 2000, has met with Barwell and Mallinder in recent days. Both requested he return to training.
But the 2003 World Cup winner was conspicuous by his absence from the Northampton squad that faced Samoa in a friendly last Sunday and has not been seen this week.
Barwell explained: "Ben came to see me and said he was unhappy about not being appointed captain. He said: 'That's it, I'm off'.
"I am bitterly disappointed both with Ben's decision and also with his action in staying away.
"He has been a brilliant servant to the club for more than 11 years but no man is bigger than the club.
"It is the director of rugby's responsibility to appoint the captain and I support Jim's choice of Bruce Reihana wholeheartedly. His decision must be final."
Cohen, who is in the middle of a testimonial year, was anxious to captain Northampton and help the club bounce straight back into the top flight.
A spokesman for Cohen said: "Ben is saddened by the way the club have behaved and hoped that their differences could be dealt with amicably and privately.
"Given the Saints' relegation to Division One, Ben was to keen to take on more responsibility and saw the challenge of the captaincy as one he would have relished for the club he has played for his whole career and cares for deeply."
Cohen has made 270 appearances for Northampton since his debut against Treorchy in 1996 and won 57 caps for England, scoring 31 tries.
In May, Cohen withdrew from England's World Cup training squad in May in order to spend more time with his wife, Abby, who is expecting twins.