The ex-England footballer launched the legal action against News International subsidiary News Group Newspapers in relation to two articles published in July 2010, Press Gazette reports.
In a High Court writ, Gascoigne claimed that the first article - headlined 'Gazza off to rehab after lager trek to pal Moat' - published enough information about the location of his clinic to encourage people to spy on him, which had interfered with his treatment.
In the second story - headed 'It's Paw Gascoigne' - the newspaper alleged that Gascoigne groped a woman in a chemist's shop.
The 43-year-old ex-Tottenham midfielder argued that his attempted intervention with Moat was evidence of a need for treatment and that the stories breached his reasonable expectation of privacy and damaged the prospects of his rehabilitation.
Gascoigne has asked the court for an injunction banning the press from publishing information about his medical conditions and treatment, including that for alcoholism.
Last month, Gascoigne's lawsuit against the News of the World moved up from the reserve list to become one of the lead cases in the ongoing phone hacking civil actions after Andy Gray settled his case with the paper.