Moratti was left disappointed by Inter's ninth-placed finish in Serie A, although he continues to believe Stramaccioni has plenty to offer.
The president has yet to decide whether to risk allowing the 37-year-old to carry on in his job next season, and speculation in Italy surrounds the possibility of Walter Mazzarri moving to the club after he confirmed his departure from second-placed finishers Napoli.
He told reporters today, according to inter.it: "We're still making decisions, nothing's been done yet.
"(There is) no chance of anything this evening, but there's no rush. We've got time."
The Milan giants missed out on European football for the first time since the 1999-2000 campaign by clocking their lowest finish in the top flight since it was expanded to feature 20 clubs in 2004.
Stramaccioni defended the team after Sunday's 5-2 home defeat to Udinese, pointing out that his side were stricken by injuries and stressing Inter should not be judged on that display.
But a dismal run of form over the closing months of the season has cast doubt over the future of the man who took charge in March 2012, initially on a caretaker basis.
When it was put to Moratti today that Stramaccioni has shown ability, if not produced results, the president said: "I agree with you. On the one hand he's a talented guy who will definitely be a success in this business, on the other we have the situation we find ourselves in. So I'm thinking about the best choice I can make."
When asked whether he was an admirer of Mazzarri, who is looking for a new club after deciding the time was right to leave Naples, Moratti offered a cautious response.
"They're all good coaches," he said, "but then they have to work at Inter, which is something different."