After weeks of negotiations, the Iceland international finally joined the Toffees in a club-record £42m deal from Swansea last week.
Koeman revealed: "When I heard from the chairman Bill, he was always so calm. He said: 'I will do the deal but we need to be patient.' And of course I understand it's business.
"It's a big transfer. You don't do these kind of transfers in two days. Sometimes it takes longer than you like.
"'It's close, it's close, it's still close…' Of course, it would have been better to have it earlier, but finally, we have the agreement and everybody is happy about that - look through the rest of the teams. Everybody's still looking to sign new players."