Negotiations collapsed and Chelsea have finally withdrawn their
offer, so Cole, 28, can go on July 1.
German midfielder Ballack, 33, is also leaving the Double
winners.
He had wanted a one-year extension to his existing deal but once
again Chelsea refused to meet his demands as the club attempt to
impose a strict new pay code.
SunSport revealed Cole held a meeting with Arsenal boss Arsene
Wenger before flying to South Africa for the World Cup.
Tottenham chief Harry Redknapp - who gave Cole his big chance at
West Ham - has also publicly stated his desire to sign the
midfielder.

Arsenal
and Spurs are
both keen on signing him, along with Manchester United, City and managerless Liverpool.
He has turned down a return to West Ham.
While Cole and Ballack will attract much interest, there is an
uncertain future ahead for other released players.
Leeds goalkeeper Casper Ankergren is one of those in exalted
company.
'I am not the gambling type but this is a gamble,' he says.
It's not the first time he has rubbed shoulders with the game's
finest but there is a darker spectre to this - what if I don't find
a club?
Like Cole, Petrov and Fabio Aurelio, Ankergren is out of
contract but he should find a club after helping guide Leeds back
into the Championship and playing in their best FA Cup run for
years.
Yet he can't help showing concern. On average, more than 500
players are released annually.
'It's difficult,' he says. 'On one hand it is a big concern, on
the other it is exciting.'
The 6ft 3in Dane, 30, lives with his partner, Carina, and their
four-year-old daughter, Victoria, in Headingley, Leeds.
'I am reasonably secure financially but that doesn't last for
ever,' he said.
'I respect (Leeds manager) Simon Grayson a lot but I want to
prove he was wrong to let me go.'
While Cole's main challenge will be to convince Arsenal to pay
him £100,000 a week, it is at the coal faces of League One and Two
where reality bites hardest.
Morecambe have released longest-serving player Wayne Curtis at
30, and Tranmere's Chris Shuker is facing a halving of his
£1,500-a-week wages to extend his career elsewhere at 28.
The players' union are relied on for support. Already they have
listed more than 400 released players on their website, advertising
their availability.
As chief executive Gordon Taylor says: 'It is a very insecure
career.'