The Paraguay striker came close to joining the Lazio only for the deal to fall through on the final day of the transfer window.
Despite also being linked with Fulham, the 29-year-old Santa Cruz remained at City.
Santa Cruz, who is on international duty with Paraguay this week, said to Rome's Corriere dello Sport: 'I had hoped to arrive in Rome, that was my dream.
Staying put: Santa Cruz's move to Lazio fell through - but the striker won't rule out a transfer in the future
'Unfortunately, the deal fell through. There has been a lot of talk of my arrival to Italy and perhaps next time it will materialise.'
Lazio themselves have not ruled out making another attempt to acquire Santa Cruz in January.
'We will see what happens in the next few months,' said Santa Cruz. 'I don't like to look too far away and I don't know what Lazio's intentions are.
'I have received many offers but I had hoped to play in Rome. I have happy in England but Italian football fascinates me and I would love to have an experience in Serie A.'
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