Do mined coins show up on blockchain explorers and can the balance of the addresses that mined the coins be seen? Somebody wants to do a deal for a bitcoin transaction but when proof of the coins is required they say that the coins will not show up on blockchain.info as they were "mined coins".

Is this really true? Will mined coins really not show up on blockchain.info for the address' balance?

If that is the case how can proof of funds of bitcoins be seen before the deal goes through?
They DO show up in the blockchain. The block reward(new mined coins) is sent to the miners address who has found a block via the coinbase transaction.

The coinbase transaction can be viewed and is found as the first transaction in every block.