Network Rail is "on the right path" towards agreeing a pay deal with staff to stop train strikes, the company's chief negotiator has told Sky News.
It comes as passengers faced more disruption from a fresh round of industrial action by the Rail, Maritime and Transport union (RMT) today.
Around half of Britain's railway lines were closed and just one in five services were running as 40,000 RMT members at Network Rail and 14 train operators stage two 48-hour walkouts - one that started on Tuesday and another which will begin on Friday.
Many places have no services, including most of Scotland and Wales. Drivers in the Aslef union will strike on Thursday.
'2,000 people need to change vote for deal to pass'
Network Rail operates Britain's railway infrastructure. Its chief negotiator Tim Shoveller told Sky News that the company wants to "make clarifications" to the RMT about the offer its members rejected and suggested it will be resubmitted and put to another vote by members.
"We think we're about 2,000 people short to secure the new deal, so we're on the right path," Mr Shoveller said.
He said: "We made really significant progress in September, when we put a new offer on the table that was worth 9% over two years and no compulsory redundancies until 2025 - and a whole load of other things the unions have been asking for.
"Two of the three unions (Unite and the Transport Salaried Staffs' Association - TSSA) accepted that, and we know with some quite high results from our employees who voted for the deal even though RMT suggested they shouldn't vote for the deal."
He added: "There's a new contract available within the maintenance organisation that actually has a much higher salary - so some people are nervous about moving on to that contract because it has different financial rates for overtime, for example. It's quite complex so we think that by explaining that better, ideally with the RMT's support, that would do the deal."
Transport Secretary Mark Harper has told Sky News "there is a fair and reasonable pay offer on the table" and said it is "deeply unhelpful" that RMT members have chosen to continue strikes this week rather than holding further discussions in an attempt to end the dispute.
But RMT boss Mick Lynch has hit out at ministers, saying that while he was available and keen to negotiate throughout the Christmas period the other side was busy "enjoying the holidays".
Mr Lynch, the union's general secretary, told Sky News: "What has happened is we had a meeting with the railway secretary on 15 December, and we'd already given notice of action, and they've done nothing since that time, over Christmas, enjoying the holidays.
"We were available to meet and they didn't want to meet us. So they've waited until the strikes come to make these noises, but if they'd wanted to get a settlement and get the strikes off, they could have taken a bit more action and got things moving."
Mr Lynch added it is the "direct responsibility" of Mr Harper that negotiations with rail companies fell through at the last minute in December.
He said: "We were talking to the train-operating companies about a potential settlement and working on documentation with them, and at the last minute the Department for Transport (DfT) - and he is the secretary of state - the department for transport intervened and insisted that driver-only operation of trains, the removal of guards from every train in Britain, was put into the documents.
"I know he did that because the people I talk to on the other side of the table have told me that the DfT have insisted that it's in there. So, that prevented any move forward on the issue."
However, Mr Harper denied that ministers intervened in rail strike negotiations in December to stop a deal, saying: "That absolutely isn't true."
He added that the "very fair pay offer on the table" has been accepted by two of the trade unions on Network Rail, as he urged the RMT union to "get off the picket line" and do the same.