The pair were dropped from the first-team squad at the weekend after being involved in a police check in the early hours of Friday morning - less than 48 hours before Saturday's 1-0 defeat at Bayer Leverkusen.
The club have now decided the pair will not feature again this term.
"We need the full concentration of every single player for this end of the season," the club's director of sport Thomas Eichin said.
"We have made it clear to Marko Arnautovic and Eljero Elia once again that their behaviour has damaged the team in this phase.
"Both will now follow individually adapted training schedules away from the first team between now and the end of the season."
With three games of the campaign remaining, Bremen are just two points above the bottom three and face 17th-placed Hoffenheim next weekend in a vital match for both sides.
Bremen have not won in their last 10 Bundesliga games and are enduring their worst season since Thomas Schaaf, the longest-serving coach in the Bundesliga, took charge nearly 14 years ago.