The latest exchanges in the row over players' rights mean that the League's general assembly on Wednesday is likely to determine whether the new season starts on Saturday and Sunday as planned.
The problem revolves around article seven of the proposed agreement relating to the protection of players who are omitted from their club's squads.
The clubs want greater flexibility but players are worried that they could be forced to train separately or accept a transfer.
"With all the problems the country is suffering, I'm a little ashamed of my citizenship," said Abete on the FIGC website.
"With the country going through so many problems, this article does not deserve so much attention."
"A little bit of good sense will allow a clear interpretation of article seven of the collective agreement," he added.
"If the contract is not signed because of the interpretation of article seven, it means something else is at play," he said.
"If there are other motives, they need to be explained."
The Italian Players' Union (AIC) held a meeting in Milan at the same time and its president Damiano Tommasi reaffirmed that strike action was on the cards.
"I reiterate our position that if the contract is signed, we will play on Saturday and Sunday, otherwise we will stay at home," he told reporters.
"I hope that Abete's interpretation satisfies the League and convinces them to put a signature on the dotted line."
The union said that the agreement was reached with the League last year after intense negotiations during which strikes were twice narrowly averted.
A last-minute deal in December meant all players with first-team contracts would be allowed to train with the main squad while rules to avoid discrimination were agreed for unwanted footballers who are put up for sale.
The AIC said they signed the same month but the League has been dragging its heels.
Tommasi added that there had never been any question that players would pay a so-called solidarity contribution" introduced by the government as part of an austerity package this month.
The measure is a temporary tax on everyone earning over 90,000 euros a month, although last week there had been reports that the players wanted the clubs to pay on their behalf.
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