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I hope LifeSiteNews is correct but I fear there is substance to this allegation.

From the National Catholic Register (not a liberal MSM source):

Pope Francis on Homosexual Unions
by Jimmy Akin Wednesday, March 20, 2013
Excerpt:
Two Situations The CDF document envisions two situations:

In the first, it speaks of when recognizing homosexual unions is first proposed in a legislative body and it says that, in that case, Catholic lawmakers must vote against such a law.
In the second, homosexual unions have already been given legal recognition and the effort is underway to try to limit their harm since it is not possible to get rid of this recognition altogether.

The Argentine Situation
In 2010, the situation Argentina fell somewhere between these two.

Unlike the first situation, this was not the first time that legal recognition was proposed for homosexual unions. As early as 2002, legal recognition had been given to civil unions in Cardinal Bergoglio's own city of Buenos Aires, and such unions had been making dramatic strides in subsequent years.
Unlike the second situation, full-blown legal recognition of the unions as marriages was not yet in force on a national basis (though various courts had already started approving them in 2009).
In that the case, it could be licit to try to stop the full-blown national recognition of homosexual marriage by tolerating a lesser form of legal recognition (i.e., civil unions) as a stopgap measure to prevent the even worse situation from happening.
After all, it would be vastly harder to roll back a homosexual marriage law once it's in force than to stop it from coming into force.
If the only way to stop it from coming into force would be to tolerate a less-bad form of legal recognition then that would seem permissible.

Cardinal Bergoglio's Proposal?
Cardinal Bergoglio thus may have entertained the idea of civil unions as a last, desperate stopgap measure to prevent full-blown homosexual marriage from being inflicted on Argentina.
He may have floated the idea to his fellow bishops, who ultimately decided that it wasn't the way to go on the issue.
He may have mentioned it privately in discussions with homosexual activists, as a kind of trial balloon, to see if they would be willing to accept it in place of homosexual marriage.
It's hard to know what happened in these situations, and I'm not going to take an activist's word about what happened in a private discussion of which we have no transcript. It's too easy for partisans to slant what may have been said--or even engage in outright fabrications.
I prefer to go with what a person has chosen to say in public, on record, in a verifiable way.

A Desperate Expedient?
It's clear that Cardinal Bergoglio in no way approves of homosexual marriage, which he attributed to the Devil. That's a remarkably blunt statement.
If he privately floated the idea of civil unions as a way of stopping full-blown homosexual marriage, I would see this as a last, desperate expedient and not suppose that he viewed such unions as a positive thing, just as a less-horrible thing.
I certainly wouldn't see it as evidence that he was "seeking compromise" or being "flexible" or open to "dialogue" on established moral principles.
I would instead assume that he was trying to prevent an even worse situation from coming about and seeking to apply to the particular situation in Argentina the principles the Magisterium had already established.