The 27-year-old is currently a free agent after leaving Championship winners Leicester City upon the expiration of his contract last month.
According to Football Insider, Iheanacho has already received offers from clubs in Greece and Turkey, but he could yet continue his career in the Premier League.
Indeed, the report adds that both Forest and Ipswich are 'racing' to sign the former Manchester City youngster and are hopeful of convincing him to stay in England.
It remains unknown as to whether Iheanacho would be interested in joining either Forest or Ipswich, but both clubs are now set to make formal contract proposals to the striker in the hope of winning the race for his signature.
Iheanacho spent seven seasons as a Leicester player and scored 61 goals in 232 appearances across all competitions, although he was only able to make the net ripple six times in 26 outings for the Foxes last season.
Adams completes Torino move after leaving Southampton
One striker who has opted against staying in the Premier League is Che Adams, who has finalised his transfer to Serie A outfit Torino.
The 28-year-old officially left Southampton after his contract expired on June 30 and the Scotland international was linked with a number of English clubs including Wolverhampton Wanderers.
Adams has decided against extending his contract at Southampton despite helping the club secure promotion back to the Premier League, scoring 16 goals in 2023-24, and will now embark on a new challenge abroad with Torino, who finished ninth in the Italian top flight last season.
The striker has taken to Instagram to address his departure from Southampton, saying: "Thank you, Southampton Football club. I'm so grateful for the last 5 years. Southampton became home to me and my family, and we were surrounded by some amazing people & players.
"Thank you to everyone, from the backroom staff, to the kitchen staff, to the coaches, for all your support & guidance since I stepped into Staplewood in 2019.
"I'm proud to be leaving with my head held high, getting this club back to where it belongs. We made you fans a promise, and it gives me such pride we delivered. Thank you for all the support, you've followed us home & away in incredible numbers, and it meant so much. Wishing you all the best for the future!"
Adams has put pen to paper on a three-year deal with Torino and becomes the club's fifth signing of the summer transfer window.
Ex-Barcelona striker agrees to join Gremio
Elsewhere, Brazilian club Gremio have announced they have reached an agreement to sign Martin Braithwaite on a free transfer.
The 33-year-old striker has been without a club since leaving Espanyol last month, departing the Spanish club after scoring 32 goals in 78 appearances over the last two seasons.
Braithwaite was strongly linked with a move to Greek giants Olympiacos earlier this month, but journalist Fabrizio Romano claims that his proposed move to the Europa Conference League winners has collapsed.
The Denmark international has now agreed to make the move to South America for the first time in his career and join Gremio, who parted ways with veteran striker Luis Suarez in January.
"Martin Braithwaite! Gremio reaches an agreement to sign the Danish centre-forward," a statement from Gremio read on X. "The athlete comes to Porto Alegre in the next few days to undergo medical examinations and sign a contract."
Braithwaite has previously played in his native Denmark, France, England and Spain, most notably representing Barcelona, albeit as a peripheral first-team figure, between 2020 and 2022.