Following the hearing in Cardiff, Judge Keyser KC said he would reserve his decision until a later date.
Mr Howells wants access to the Docksway landfill site, where he claims a digital wallet was mistakenly dumped by his former partner.
The hearing was to determine whether the case should go to a full trial.
Mr Howells has argued that his former partner erroneously dumped the hard drive containing a Bitcoin wallet in 2013.
He was an early adopter of cryptocurrencies and successfully created, or “mined”, the Bitcoin when it was worth a small fraction of its current value.
After Mr Howells launched legal proceedings, the council applied for a High Court hearing to ask a judge to dismiss the claim before going to trial.
