The 24-year-old has been strongly linked with a move to the Premier League this summer, with the likes of Arsenal, Manchester City, Newcastle United and Tottenham Hotspur all credited with an interest.
However, The Athletic reports that West Ham are now working on a deal to sign Paqueta and talks are advancing over a fee a personal terms.
Sky Sports News, meanwhile, claims that the Brazilian's agent is in London and is keen for his client to complete a switch to the Hammers before the summer transfer window shuts next week.
After a brief spell with AC Milan, Paqueta joined Lyon in 2020 and has since made a total of 80 appearances across all competitions, scoring 21 goals and providing 14 assists.
While Lyon struggled domestically last season - finishing eighth in Ligue 1 and 25 points behind champions Paris Saint-Germain - Paqueta was a standout performer for the French side, recording 11 goals and seven assists in 44 games.
The attacking midfielder, who has three years remaining on his Lyon contract, played against West Ham in both legs of their Europa League quarter-final tie, which saw the Hammers progress 4-1 on aggregate.
Paqueta, who has been capped 33 times by Brazil, could become the eighth signing made by David Moyes this summer should he complete a big-money move to the London Stadium before the September 1 deadline.
The Hammers completed the signing of left-back Emerson Palmieri from Chelsea on Tuesday in a deal worth £15m.
Nayef Aguerd, Alphonse Areola, Flynn Downes, Gianluca Scamacca, Maxwel Cornet and Thilo Kehrer have also arrived at the club on permanent deals.
Paqueta is not the only midfielder who is attracting interest from West Ham, as they are also in talks with Club Brugge over a deal to sign Hans Vanaken.
The 30-year-old has spent his entire club career in his native Belgium and has publicly expressed his desire to join the Hammers this summer.
"Clearly there is interest, enough has been written about it. New talks tomorrow and we'll see where it ends," Vanaken told reporters last weekend.
"I think it could be my last chance and I want to do it. There's still a lot to talk about, let's wait and see."
Moyes has confirmed, via journalist Roshane Thomas, that West Ham have "made an improved bid" for the Belgium international, believed to be worth €11m (£9.3m) plus add-ons.
Vanaken, who has three years remaining on his contract, has scored 102 goals and registered 73 assists in 344 appearances during his seven-year spell with Brugge.