Appointing a director of rugby - former Wasps boss Warren Gatland is a rumoured target - will be considered, but current head coach Paul Grayson's job appears safe.
Saints' fellow relegation candidates Worcester avoided the drop despite losing their first eight Premiership games, as a 22-7 victory over Saracens sealed Northampton's fate despite Grayson seeing his team edge out London Irish 27-22.
Barwell said: "I would say to the fans I am very sorry we have let you down. When I walked around the pitch after the game, the supporters were all telling me not to worry and that they would be here next season.
"As far as rebuilding the club goes, I don't think we will have any problems.
"Of course, I could have done better with the signings. You don't have to be a genius to work out that if you only win six of 22 games, then obviously you have messed up along the road.
"I blame myself. I don't wear boots and run on to the pitch, but I am the guy who picks the coaching team, so I will take responsibility for that.
"A director of rugby coming in is one of the options. Almost certainly, Paul (Grayson) will stay.
"There are lots of options, but what there won't be is knee-jerk reactions. We need to make sure we can win National League One next season and make sure we have a good squad that can immediately survive and do well in the Premiership.
"The process will be a long way along the road by the end of Monday in terms of what structure we will have in place next season."