The 22-year-old is now contracted in Alsace until the summer of 2029, with reports in France suggesting Strasbourg would want a fee in the region of €35m to consider letting him go.
The Paraguay international impressed further this summer at the World Cup, scoring in his country's memorable win over Germany during their run to the last 16 as part of a campaign that saw him feature in six matches for Rudi Garcia's side.
According to transfer journalist Ben Jacobs, Enciso is now closing in on a return to Ipswich Town, who he already knows well from a loan spell in the second half of the 2024-25 season, when he scored twice and set up three more in 13 Premier League appearances during Kieran McKenna's time in charge.
O'Neil took over at Portman Road this summer after leaving Strasbourg, and the working relationship the pair built in France appears to be a significant factor behind the growing interest.
Ipswich have already been active in the transfer market this summer and would need to move quickly if they want to beat off competition, with Galatasaray also credited with an interest in the versatile forward, who can operate through the middle or from the left.
Ipswich have spent close to £85m on new arrivals since their promotion back to the Premier League was confirmed, with five separate signings completed since July 21 alone, yet O'Neil has made clear he still wants further additions before the transfer window closes.
A reunion with Enciso would give Ipswich a player already familiar with English football's top flight, having previously operated as a No 10 under McKenna during his loan spell, in contrast to the wider, more advanced role he has since grown into at Strasbourg — a club with strong ties to Chelsea through the BlueCo ownership model that has also sent the likes of Ishe Samuels-Smith and Mathis Amougou across from Stamford Bridge in recent seasons.