Spurs suffered a 3-1 loss to Chris Wilder's side on Thursday and Arsenal posted highlights of their FA Cup victory over the Blades on Facebook with the caption: "It's not easy beating Sheffield United at Bramall Lane."
"I think if they were top of the league or fighting for top four in a really good moment they wouldn't enjoy the problems of others," he said.
Arsenal are in seventh place in the Premier League table, one point above Spurs. The two sides will play each other in the north London derby next Sunday.
Mourinho continued: "You only enjoy the problems of others when you are also in trouble. In the end it says more about them, they don't have much to celebrate, they have to get every opportunity to do it.
"They are in a very similar situation to us in the table. I don't like to connect the club with some post or tweet. Maybe I am wrong, maybe I am right, but the person that did it probably did it by themselves.
"I don't believe it was [Mikel] Arteta that posted, I don't believe it was [Granit] Xhaka or another captain that did it.
"It was probably some guy that was at home for three months working from home. No problem at home, but we will be waiting for them."
Mourinho said he was "disturbed" and "destroyed" at his side's lack of desire at Bramall Lane last week in a defeat that dented their hopes of qualifying for the Champions League.