The Marseille midfielder is set to complete a move to Craven Cottage for a fee that could reach ÂŁ30m but West Ham bosses expressed an interest at the last minute, according to Lâ€™Equipe.
Fulham successfully submitted a deal sheet for Anguissa, 22, after failing to get everything signed off by the 5pm deadline.
He will be handed a five-year contract by the Premier League new boys.
West Ham – on the hunt for a defensive midfielder – swooped for former Aston Villa midfielder Carlos Sanchez.
â€śIâ€™m very happy to be here. Iâ€™m very excited and looking forward to achieving lots,” said Sanchez
â€śI decided to come here because, first, it is the Premier League, second, itâ€™s a beautiful club and the fans are always helping the club and, third, for me itâ€™s an amazing opportunity and an amazing project the club are leading, being ambitious and keen to achieve good things.
â€śIâ€™d like to thank all of the people, the whole club, for having me.â€ť
The Colombia international has put pen to paper on a two-year deal.
The 18-year-oldâ€™s switch was announced after the 5pm deadline.
Quina has agreed a four-year deal, with the option for a further season.
A statement from the Hammers read: â€śWest Ham United wish the teenager well in his future career.â€ť