The 31-year-old Portugal international, who has won six Premier League titles and the Champions League during a nine-year spell at the Etihad Stadium, will depart as a free agent when his contract expires after the campaign concludes.
"Every good story comes to an end," Lijnders told reporters after City's 4-0 FA Cup quarter-final victory over Liverpool. "I hope he enjoys the last months - there are only six weeks left - and has a good farewell. He deserves all that attention."
Pep Guardiola, who was serving a touchline suspension during the match, has previously described Silva as "irreplaceable".
Silva joined City from AS Monaco in 2017 for a reported fee of about £43.5million and has since made 450 appearances for the club. Known for his tactical versatility, superb technique and tireless work rate, the midfielder has been a cornerstone of City's side under Guardiola.
After winning the League Cup last month, City remain in contention for a domestic treble as the 2025/26 campaign enters its final weeks, despite trailing Premier League leaders Arsenal by nine points. The Manchester club have a game in hand and eight matches remaining to bridge the deficit.