Sky Sport Italia claim the current Premier League leaders, whose manager is Claudio Ranieri, have sent an initial proposal today.
The bid is believed to be €10.5m with a three-term contract for the player worth €2.2m per year plus bonuses.
Eder was on target in the 4-2 defeat to Napoli, his 12th Serie A goal in 19 games this season.
Born in Brazil, but spending much of his career in Italy and playing his international football for the Azzurri, the 29-year-old is valued at around €15m.
Inter have also been heavily linked with Eder, but were not prepared to pay that sum.
It’s possible Leicester will aim to splash the cash so they can stay near the top of the Premier League.
Foxes boss Claudio Ranieri has expressed his desire to add another forward to his squad this month, after Andrej Kramaric left the club on loan for Hoffenheim.
However, the surprise Premier League leaders have now made an offer of £8m for Eder.
The 29-year-old is currently second in the Serie A goalscoring charts after finding the net 12 times this season.
Inter Milan are also keen on snapping the Italy international up, but they are likely to be outbid by Leicester.
The Foxes are three points clear atop of the Premier League table following the 3-0 demolition of Stoke City on Saturday, but recently offloaded Andrej Kramarić on loan to Hoffenheim for the rest of the season.
Sky Sports Italia report Leicester are prepared to meet Sampdoria’s valuation for the 29-year-old, who has scored twice in eight appearances for the Azzurri.
Éder has netted 40 times in Serie A since joining the Italian outfit in 2012 and has reportedly been offered a financial package which includes a weekly wage of £29,000-a-week.
Ranieri said this weekend: “I’d like another striker to give support to the other three [Jamie Vardy, Shinji Okazaki, Leonardo Ulloa], because our strength is playing at a high intensity”.
“It’s not possible to play at high intensity for all the season. Everyone is reinforcing or taking players because they want to save their season or want to win something.”