Representatives from the Italian club sanctioned first contact with Gerrard in October before Frank de Boer was sacked as their manager.
Initial discussions related to the possibility of the 36-year-old's playing career extending by a further six months until the end of the 2016/17 campaign.
Though Inter have struggled in Serie A this season and currently lie in ninth position, some 13 points behind leaders Juventus ahead of the Milanese derby this weekend, the idea of potentially retiring at one of Europe's grand football institutions might appeal to Gerrard.
Gerrard is currently considering several playing and coaching proposals both in England and abroad after it was announced earlier this week that his contract with Los Angeles Galaxy would not continue beyond January.
Inter appointed Stefano Pioli as their new head coach on 8 November and while it remains to be seen whether they will make Gerrard a firm offer under his guidance, sources in Italy told The Independent that the club's desire to sign the midfielder originated at boardroom level.
There is disappointment at the San Siro that Inter's big summer midfield signings have failed - so far - to consistently meet expected standards.
Portugal's European Championship winner João Mário was recruited from Sporting Lisbon for around £40m while Éver Banega came from Sevilla and neither have since been able to rise above Inter's troubled start to the season, which has witnessed defeat by Southampton and the imminent possibility of Europa League elimination.
Other options currently available to Pioli include Geoffrey Kondogbia, Felipe Melo and Gary Medel, formerly of Cardiff City. It is believed that attempts will be made in January to sell both Melo and Medel, thus freeing up squad space in their midfield department.
Meanwhile, Gerrard's negotiations with MK Dons over the League One club's vacant managerial position are at a formative stage.
His ultimate ambition involves a return to Liverpool but the challenge of establishing a role that meets the needs of the club and the ambitions of the individual concerned remains to be solved.
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