The Gemara in Sota 38b says that birds recognize when the bait for them is put out with a stingy heart, and they won't come.
The Gemara Bava Basra 16b describes Iyov's friends as being aware of each other's pain, although they were far from each other, through a crown or tree that they all had and that would wither if one was in trouble.
The Maharal explains that this is describing how they were completely united. This suggests that it was some form of telepathy that they were tuned in to each other enough to be able to sense remotely when something was wrong. People in our times have had such experiences, but not reliably.
Then we have the famous Pasuk in Mishlei(see Ralbag) telling us that the way you feel about a friend he will feel about you.
There is also the Gemara in Sanhedrin About Yoav finding out about king David being in trouble.
These aren't conclusive but are easier understood with the concept of some form of telepathy. This won't prove anything to someone who didn't already believe in it but can serve as confirmation.