Both clubs have agreed a fee with Blackburn and personal terms with the versatile Scotland international.
Dailly, who knows he has no future at Ewood Park, is talking the situation over with his family before deciding on his next move.
Taylor confirmed today that Leicester have had a £1.75m bid, rising to £2m on appearances, accepted by the First Division club.
The Foxes boss believes Dailly's links with the East Midlands - he left Derby for Blackburn in a £5.3m deal in 1998 - will tip things in the Filbert Street club's favour.
Dailly remains firm friends with Leicester pair Gary Rowett and Callum Davidson, his Scotland room-mate.
Taylor said: "I know Christian has spoken to West Ham but I am quietly confident he will come here.
"He knows the area and knows one or two of our players. They've all said what a good pro he is.
"I need another central defender and he fits the bill.
"He is very experienced. I know he can play in midfield but I see him as a defender here."
Hammers boss Harry Redknapp admitted it would be "a sickener" to lose out on Dailly after he "thought I had him in the bag".
But Dailly said: "It's sometimes difficult to be patient in football because people want things done in 10 seconds.
"But that's not my style. I like to take my time, and it's not as if I've asked for a move.
"I went to talk to West Ham and have a look around, and everyone I spoke to impressed me.
"It makes a decision more difficult to make when you've got a family to think about.
"I'm 27 so this is a big move for me because whichever club I move to are going to get my best years.
"And if I sign for them for the next four or five years then it's got to be right."
Dailly has made just 12 appearances for the first team this season and does not seem to be part of Graeme Souness' long-term plans.