The Rail and Maritime Union has drawn up a list of demands following the terrorist attacks on the captial's transport network, and has said it is willing to ballot members on strike action if they are not met.
They include more guards on trains, a "no radio, no train" rule for trains with faulty communication systems and stronger drivers' cabs.
Today's talks, which are expected to last all afternoon, will then go back to a meeting of the RMT executive on Friday morning, where a decision will be taken on balloting members on possible strike action. The union has around 11,000 members working on the underground.
Bob Crow, the general secretary of the RMT, told the BBC ahead of the meeting: "This gives the company another opportunity to demonstrate that it will put in place additional measures that we have been seeking to improve the security.
"Staff in the front line have acted selflessly and professionally following the recent bombings and they have serious concerns and proposals that deserve serious consideration."
The union has drawn up a nine-point wishlist of demands:
· no weakening of safety guidance rules for deep-stations, such as King's Cross, which were brought in in the wake of the 1987 fire there
· trains without reliable radios to be taken out of service immediately
· clarification of the existing "code amber" procedures - whether or not to close the entire network after suspect devices or incidents occur on one line
· no reduction in uniformed staff
· a review of the current system for dealing with suspect packages, first introduced in the 1970s do deal with the IRA, known as "HOT" - hidden, obvious or typical.
· a second driver or guard to be onboard all trains to help with emergencies
· better training for front line staff
· additional equipment for staff, such as breathing apparatus
· consideration of strengthening driver cab bulkheads
Meanwhile, a London Underground report out today shows that promised improvements by the tube's two private maintenance companies are behind schedule.
The companies - Metronet and Tube Lines - have been responsible for continued signal failures and engineering overruns have increased by 35% year on year, the report says.
London Underground managing director Tim O'Toole said: "In the days following the July 7 terrorist attacks, both Metronet and Tube Lines have performed well in the planning and support they have given to LU as we have worked to restore services through the affected sites at Aldgate, Edgware Road and Russell Square. We need to see this focus and application on a consistent and ongoing basis."
"We are paying a premium price for improvements, but performance to date has not been good enough."
