Praise be to Allah.
It was proven from Abu Ayyoob (may Allaah be pleased with him) that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Whoever fasts Ramadaan then follows it with six days of Shawwaal, that is the fasting of a lifetime.” Narrated by Ahmad, 5/417; Muslim, 2/822; Abu Dawood, 2433; al-Tirmidhi, 1164.
This is a saheeh hadeeth which indicates that fasting six days of Shawwaal is Sunnah. Al-Shaafa’i, Ahmad and a number of leading scholars followed this. It is not correct to oppose this hadeeth with the reasons that some scholars cite for this fast being makrooh, such as the fear that an ignorant person might think that this is part of Ramadaan, or the fear that people may think this is obligatory, or that he did not hear that one of the earlier scholars used to fast these days, because this is all speculation, which cannot be used to oppose the saheeh Sunnah. If something is known, this is evidence against those who do not know.
And Allaah is the Source of strength.