Hopes were rising today that talks would soon resume on settling the Tube strike that caused severe difficulties for London commuters for a second morning.

Transport for London said that it was waiting to see draft proposals sent over by the RMT union last night to resolve the dispute. The final sticking point is the threat of compulsory redundancies among 1,000 backroom Tube staff whose jobs are under threat following the failed part-privatisation.

"We welcome this development and look forward to receiving the draft this morning and we are preparing for further discussions at Acas [the conciliation service] as soon as possible," a spokesman for TfL said.

London Underground ran 160 trains an hour during the peak today — about one third of the 500 trains an hour usual at that time. Yesterday it operated 130.

Five of the 11 Underground lines were offering a partial service during the rush hour, and the Northern Line, while very busy, was operating almost as normal. "Where we can we will try to improve on that during the day," said a TfL spokesman. "The issue for us is the maintenance of trains, and how long they can continue to run without being looked at."

By 10am the number of lines in full or partial service had risen to eight, with only the Hammersmith & City, Waterloo & City and Circle lines still entirely suspended, and the Jubilee running almost as normal with some delays. Trams, overground trains and the Docklands Light Railway were operating with some delays.

The worst effects of the strike were being felt by drivers, with a second day of gridlock on the approach roads into Central London. Travellers told of spending three hours in traffic to go eight miles.

Bus passengers have also endured long waits to board overcrowded buses, despite an extra 100 vehicles being put into service.

The 48-hour strike, which has been only patchily upheld by RMT members, is due to end at 7pm.

Boris Johnson, the Mayor of London, refused to accept any blame for failing to avert the walkout as he talked to commuters using a shared taxi initiative at London Bridge station this morning, one of numerous emergency measures to help Londoners get to work.

"Once again I apologise for the disruption but I must say I lay the blame squarely at the door of the extremely ill-advised leadership in the RMT," said Mr Johnson.

He has been criticised for failing to negotiate with the union in person — including unhooking his earpiece and refusing to take part in the recording of an interview on Channel 4 News last night after learning that he would be expected to appear alongside Bob Crow, the RMT leader. He was interviewed separately later instead.

Ken Livingstone, the former Mayor, said that Mr Johnson's stance — refusing to take part personally in face-to-face talks with Mr Crow so long as there was a strike threat hanging over the capital — was a mistake.

Writing in The Times today, he added: "Although Boris Johnson gave an election pledge to introduce a no-strike deal, we know that not only has he failed to meet the transport union leaders in the 14 months since his election but he did not even bother to send a letter asking them to consider a no-strike deal," said Mr Livingstone.

Mr Crow said he was "ready and willing" to resume talks today. The RMT said it had made four attempts yesterday to invite TfL back to the negotiating table at Acas yesterday, including a direct written appeal to Mr Johnson which it claimed was turned down.

Meanwhile, more than 58,000 England fans managed to get to Wembley last night to watch England beat Andorra 6-0 in a World Cup qualifier, but the Football Association still faces an estimated £1 million bill to keep its promise to refund about 12,000 supporters who had tickets for the game but could not attend because of the travel difficulties.

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