Johnson is set to join The Owls for around £400,000 from the West Yorkshire club after becoming a firm favourite at the newly-named Intersonic Stadium, formally Valley Parade.
A £250,000 bid from Derby County was turned down and it is believed that Wednesday's offer is well in excess of that.
Wednesday must complete the deal for the Jamaican soon as the transfer deadline ends tomorrow night - when Brian Laws and his team will be on their way back from Southampton.
Alas, the flying winger will not be able to play for Wednesday straight away after getting sent off against Yeovil and will sit out two games for his new club.
The former Oldham and Bolton star has scored four goals this season from 26 league starts.