Pre-tournament favourite France topped a tough group containing Portugal, Germany and Hungary, only to be eliminated at the last-16 stage on penalties by Switzerland.
It marked the first time since the 2010 World Cup that Les Blues had failed to reach at least the quarter-finals of a major competition.
France's premature exit led to questions about Deschamps' future, with Zinedine Zidane being heavily linked with the job since leaving his position as Real Madrid boss in May.
Deschamps is under contract until after the 2022 World Cup and has been in charge of his country since 2012, lifting the World Cup in 2018 and reaching the Euro 2016 final in that time.
La Great insisted last week that the 52-year-old's position is not under immediate threat, but the FFF president plans to meet with Deschamps in the coming days.
"We will meet next week. I always let 10 days pass first," La Graet said. "We'll meet in my office in Guingamp and spend the day together.
"Everyone has to think about what went right and what went wrong. We'll spend the day chatting. He's a friend, someone who has always been faithful. This is his first failure."
When asked about his recent comments of Deschamps' job being safe, La Graet said: "I didn't say that. I said we will have a discussion. We will speak about what went wrong."
France beat Germany 1-0 in its opening Euro 2020 game, but it then played out draws with Portugal and Hungary before losing on penalties to Switzerland following a 3-3 draw.
The world champion squandered a commanding 3-1 lead against the Swiss in Bucharest and was knocked out 5-4 in the shoot-out.
Les Blues have now been eliminated in their last three games in which they played extra time in major tournaments, as many as in their first 11.
Reflecting on his nation's shock exit in the first knockout round, La Graet said: "I feel disappointment more than anger. We must remember we had a difficult group.
"We finished top of our pool, and switched off for 10 minutes against Switzerland. Not everyone played at the same level, but we can't throw away everything we've previously done.
"We have quality players and coaches. We will analyse things that didn't work next week. Sometimes it's small details."
France is back in action on 2 September (AEST) with a FIFA World Cup qualifier against Bosnia and Herzegovina.