1) This is in the Perush ha'Rosh on 77b, end of DH (v')Amar Rebbi Yochanan, three lines from the bottom of the page. The Rosh writes in the name of Rabeinu Eliezer of Mitz that Rebbi Yochanan said that he must do Bitul in his heart. This means that he must say in his heart, "It is annuled for you," but he must not say this with his lips loudly.

We learn from the Rosh that he does say, "Mufer Leich," but he does not say it loudly with his lips. This implies that he does say it with his lips, but not loudly. The reason that he should not say it loudly is so that other people should not hear it, but he himself does hear it.

2) I was looking at this question again and I see what is disturbing you.

The problem is that the Rosh (end of 77b) writes in the name of Rabeinu Eliezer of Mitz that he must say in his heart, "Mufar Leich," but he should not say this loudly with his lips. The Rosh compares this to the Gemara in Berachos 15a ("15b" that is written there is a printing error), where the Gemara states, "One should not say Birkas ha'Mazon in his heart, but if he did say the Berachah this way, he is Yotzei." The Rosh explains that he made the blessing but the blessing was not "heard by his ear." (This is also what Rashi writes there (DH b'Libo) -- that when the Gemara states that he should not say the Berachah in his heart, this means that "his ear did not hear it.")

This is not consistent with what we wrote earlier -- that he said it so quietly that only he can hear it -- since the Rosh writes that his ear did not hear it.

3) The Rosh in the back of the Gemara (10:9) also cites Rabeinu Eliezer mi'Mitz. The Korban Nesanel (#3) writes that Rabeinu Eliezer mi'Mitz maintains that it does not help what he thought in his heart, even if he thought "Mufar Lechi." Rather, he must verbalize it with his lips quietly. It seems that this means that he moves his lips in the same way that he would move them if he was saying the words, but he does not make any sound, since the Rosh writes that his ear hears nothing.

4) According to this, the implication that I made in my first reply -- that the fact that Rabeinu Eliezer mi'Mitz writes that he did not say it loudly implies that he said it quietly but he could hear it himself -- is not correct. The correct Pshat would be that he moved his lips but did not even hear it himself.

5) I found, bs'd, that the Chidushei Chasam Sofer (Nedarim 77a, end of DH Mafirin) is a proof for what we wrote. The Chasam Sofer writes that the "Hirhur" ("thinking") mentioned here concerning doing Hafarah in one's heart really means a weak speech, similar to the thought of Pigul.

a) The Chasam Sofer is referring to what Tosfos writes in a few places in Shas (including Bava Metzia 43b, DH ha'Choshev), that the thought required for Pigul (when the Cohen had intention at the time of sacrificing the Korban that he would eat it at the wrong time according) is actually more than mere thinking; it in fact requires the Kohen to say that he has the wrong intentions. We see from Tosfos that even though something is called Machshavah by the Gemara, it can mean that in reality one is required to speak.

b) The Chasam Sofer adds a Chidush -- that when Tosfos writes "Dibur" this means weak speech. In a footnote to the Chasam Sofer, it is pointed out that in fact the Rosh does not mention anything about Pigul, but rather the Rosh compared this to Birkas ha'Mazon said "in one's heart."

c) We can also add that the Rosh writes that saying Birkas ha'Mazon in one's heart means that he did not hear it with his ear. Even so, the Chasam Sofer called it weak speech, which seems to suggest that he did hear it himself. It appears that the essential part of the Rosh's comparison to Birkas ha'Mazon is that nobody else heard it, even though he himself did hear his own soft speech. This is a support for what we wrote, that he says it so quietly that only he can hear it.

Kol Tuv,

Dovid Bloom