Currently, you can prove ownership of coins by signing with
the private key of an address where coins are stored.
Is there a ZKP (Zero Knowledge Proof) way of demonstrating
that you own at least X BTC ?
Here is a simple idea that is not completely ZK: find a bunch of coins that contain roughly the same amount as the coin you want to prove ownership of. Then use an Abe-Ohkubo-Suzuki ring signature[1] to prove that you own one of the private keys. The more coins you pick the less information you leak but the larger your signature get.
I guess that there is a ZKP, but checking it requires to process the full UTXO set and probably the size of the signature would be similar in size as the UTXO set.
[1] Masayuki Abe, Miyako Ohkubo, and Koutarou Suzuki, 1-out-of-n signatures from a variety of keys, In Advances in Cryptology - ASIACRYPT 2002, LNCS, Springer-Verlag, 2002, pp. 415432.