Latest "load of cabbage" leaves Rafa in limbo
The holders, who had already qualified for the last 16 albeit unconvincingly, were embarrassed 3-0 by eliminated Werder and now head to the Club World Cup in the UAE desperate for success to help the Spaniard cling to his job.
"I don't want to create problems before or during the World Club Cup. Afterwards we'll see," Moratti told reporters on Wednesday.
"Benitez must have character because if not, we won't win the World Club Cup."
Inter, who have lost seven games already in all competitions this term, face a semi-final on December 15 before a potential final with Brazil's Internacional on December 18.
A nervous and disjointed Inter look a completely different side from the Jose Mourinho vintage of 2009/10, which won the Serie A title, Champions League and Italian Cup and lost just seven games all season.
Injuries have robbed Benitez of a consistent line-up and formation while the Spaniard also fielded a second string side in Germany with qualification assured.
Last season's top scorer Diego Milito could return for the Club World Cup after over a month out injured but even the Argentine looked off form earlier in the campaign and the purchase of a striker in January looks certain.
The Nerazzurri are fifth in Serie A after a 3-1 weekend defeat at Lazio left them 10 points behind leaders AC Milan and making a sixth successive scudetto start to look very unlikely.
Benitez struggled in his final season at English Premier League club Liverpool and his troubles have followed him to Italy.
Moratti had previously said he would have axed Benitez already 10 years ago when he was more trigger-happy but wins against Twente Enschede and Chievo before the last two defeats bought the coach some time.
The sunshine of Abu Dhabi at the World Club Cup may be a welcome break from Milan's wintry temperatures, but for Benitez the heat is very much on.