Aslef has also demanded promises on job security including no compulsory redundancies.
The two sides are meeting at the offices of Acas, the national conciliation service, and have until 2000GMT to resolve their differences.
The train drivers' union Aslef has promised a repeat of Monday's tube strike which crippled London's transport service if there is no resolution of the dispute.
But Aslef general secretary Mike Rix said there "could be light at the end of the tunnel" in the dispute over London Underground's proposal to part-privatise the tube.
'Constructive talks'
He described talks he had with London Transport Commissioner Bob Kiley on Wednesday as "constructive".
Aslef is hoping that Mr Kiley can broker a formula that will head off any more strike action.
The two sides had an 11-hour session at Acas on Tuesday.
The union believes that private sector takeover of the tube lines will compromise safety on the underground.
Aslef has also demanded promises on job security including no compulsory redundancies.
It says its members will strike on successive Mondays, 12 and 19 February, if their demands are not met.
Last Monday's dispute closed several lines and brought gridlock to central London.
Meanwhile, the Rail Maritime and Transport union will hear from the Court of Appeal on Thursday whether its bid to overturn a ruling that their strike ballot was invalid has been successful.
Despite being ruled invalid, many RMT workers came out in sympathy with their Aslef colleagues last Monday.