Two men want Google to stop linking their names to internet information referring to criminal convictions.
They say the convictions are more than a decade old and legally "spent".
Both have taken legal action against Google bosses, arguing that they have a "right to be forgotten".
A judge is due to consider rival arguments at trials later this year.
Mr Justice Nicklin oversaw a preliminary hearing at the High Court in London on Thursday.
He said the cases were the first of their kind to be aired in England.
The judge made an order barring the men from being identified in media reports.
He said issues surrounding reporting would be analysed at a hearing in the next few weeks.