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.
Comments: Our rules
We want our comments to be a lively and valuable part of our community - a place where readers can debate and engage with the most important local issues. The ability to comment on our stories is a privilege, not a right, however, and that privilege may be withdrawn if it is abused or misused.
Please report any comments that break our rules.
Read the rules hereComments are closed on this article