Forward Ayew follows in the footsteps of brother Andre, who scored 12 goals in 35 appearances for Swansea before leaving to join West Ham for £20.5million ahead of the 2016-17 season.
"I'm sure his brother has given him the inside track on the club," said Swansea head coach Paul Clement.
"He is a very good acquisition. He's versatile and dynamic to attack the lines and get in behind defences.
"He will certainly give us something different to what we've got."
Taylor, meanwhile, brings to an end a six-and-a-half-year spell at the Liberty Stadium, having joined the club from Wrexham in 2010.
An ankle injury led to Taylor losing his first-team place to Ben Davies during Swansea's second season in the top flight, but the 27-year-old returned to the fold after his Wales team-mate left for Tottenham in 2014.
"Neil has been a great servant for this club," Clement added. "I had a really nice discussion with him about his position.
"He said he wanted to take this opportunity and I supported that. It's the right time for him to have a new challenge and we wish him all the best."
Taylor made 179 competitive appearances for Swansea and has won 37 Wales caps.