He also said that Rav Soloveitchik was enthralled by the Baal HaTanya's profoundity. He related that Rav Solveitchik gave a daily 5 hour shiur in the summer after his wife was niftar. At the conclusion of this he insisted on giving a shiur in Tanya. He also has extensive writings on the Tanya.
A similar assertion regarding the meeting is reported in Wikipedia:
According to Chabad tradition, Rabbi Shneur Zalman of Liadi and Rabbi Menachem Mendel Horodoker were sent to the Vilna Gaon by the Maggid of Mezeritch and the Gaon refused to meet with them. Rabbi Leibel Shapiro, the current Rosh Yeshiva of Tomchei Tmimim Miami, has said that at a Yud-Tes Kislev farbrengen in Boston, Rabbi Joseph Ber Soloveitchik described this event the way it was passed down through the Brisk dynasty (Rabbi Chaim Volozhin the scion of the Brisker dynasty was the prime student of the Vilna Gaon) and in this version, the Alter Rebbe was accompanied by Rabbi Levi Yitzchok of Berditchev. Rabbi Soloveitchik said that the Gaon's reason for not meeting with the Hassidic Rebbes was that he saw the holy features of the Alter Rebbe's face and realized that if he let him in "after two hours he would leave the room and join them in spreading chassidus".[2]
However R' Eliach in his sefer on the Gra page 907-912 concerning the reason for not meeting with the Baal HaTanya does not mention such a view. There is also a story - which he rejects - that claims that the Gra's mother prevented her son from meeting with the Baal HaTanya. R' Eliach does cite the Brisker Rav [page 910 note 57] who stated that the Gra didn't meet with him because he thought it was a waste of time becaue they had irreconcilable views in hashkofa. There is in fact a letter from the Baal HaTanya to his chasdim in Vilna telling them to not waste time on debates because of the differences in hashkofa are irreconcilable.
It is also clear that the Gra considered them kofrim - and simply didn't want to argue with apikorsim. These views makes much more sense.
Thus this assertion that the Gra was afraid of being influenced by the Baal HaTanya is not a tradition amongst the Chassidim, is not mentioned in the letter of the Baal HaTanya dealing with the failed meeting and was not mentioned by the Brisker Rav, and is apparently unknown in the extensive literature on the subject - including R' Eliach