Balogun’s proposed £40M switch from AS Monaco to Merseyside collapsed at the last minute after the USMNT international changed his mind and withdrew from the negotiations.
Sources close to the player claimed Balogun was unhappy with how Everton handled his medical, and the subsequent transfer talks, saying the former Arsenal man did not feel comfortable proceeding with an agreement.
The club have since hinted at their willingness to sell him in the January market - with Balogun open to moving on - but Sheringham believes it won't be to join David Moyes.
"Balogun at Everton? It would've been tough for him, if you look at what's expected as a centre forward at Everton," as per the Liverpool Echo.
"They've had forwards who have come to the fore now and again, like Beto and Barry.
"It's a tough place to be a striker. You have to score goals and work your socks off, and then you can be a hero. That's hard when you have to play up front on your own."