The meeting, which apparently took place at Luis Enrique's home, prompted speculation Barca were preparing to appoint the former Spain midfielder to replace Martino, who has had a disappointing first season in charge at the Nou Camp.
"Nobody needs to ask my forgiveness," Martino, who has contract until the end of next term, told a news conference ahead of Sunday's La Liga game at Elche.
"Nothing the club has done is inappropriate," added the Argentine. "They did not make any error. Either now or in the 10 months I have been here."
At the start of last month, Barca were on course to repeat their historic treble from 2009 when they won the Champions League, La Liga and the King's Cup under Pep Guardiola, becoming the only Spanish club to achieve the feat.
However, they faltered in key games, losing narrowly to Atletico Madrid in the quarter-finals of Europe's elite club competition and Real Madrid in the Cup final.
Their bid for a fifth domestic league title in six years also appeared to be over but Atletico's defeat at Levante last weekend and consecutive draws for Real reignited their campaign.
With two games left, Atletico are top on 88 points, Barca have 85 in second and Real 84 in third and the top two meet at Barca's Nou Camp stadium on the final day of the season next weekend.
If Barca win their remaining two fixtures they will be champions.
"We are happy to have been mistaken in giving the league up for lost," Martino said.
"Anyone would have done the same but we are pleased to have a chance again.
"If we are in this situation it is because we deserve to be, just like the other teams who are in the mix.
"The ups and downs of the league this season were not a surprise because that's football and anything can happen."
(Reporting by Iain Rogers, editing by Justin Palmer)
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