None of the band members were in court for the start of the two-day preliminary hearing on Wednesday.
But lawyers for guitarist Summers and drummer Copeland are asking the court for permission to bring new and additional arguments that they are owed money from all downloads and streaming income, under the terms of their prior agreements made in 1997 and 2016.
They say the language of those agreements should be interpreted in light of changes to the music industry, where streaming income has largely replaced the sale of vinyl, CDs and cassettes.
In court papers, they acknowledged a recent payment from Sting and his publishing company, disclosing a figure of $870,000 (£647,000), but noted that no interest had been added to the "historic underpayment".
Speaking in court on Wednesday, the musicians' lawyers said the total value of their claim was "not less than £8m" and "will be considerably larger" if their amended case is allowed to proceed.
Meanwhile, Sting's lawyers argue that he shouldn't have to give his former bandmates royalties when The Police's music is played on streaming services like Spotify because that counts as "public performance" rather than a sale.
Furthermore, they say Summers and Copeland are not eligible for royalties from streaming and digital sales because their 2016 agreement only allows for royalty payments derived "from the manufacture of records".
They added that the musicians' attempt to amend their case should be thrown out as it has "no real prospect of success".
The hearing, before Mr Justice Bright, is due to conclude on Thursday, with a trial expected at a later date.