This viewpoint is from Fatwa al-Lajnah al-Daa’imah, 10/391.
Muslims can start fasting six days of Shawal from the second day of Shawal, because it is haram to fast on the day of Eid (Eid ul-Fitr). You can fast the six days at any time during Shawal, although the best of good deeds are those which are done soonest.
These six days don’t need to be fasted immediately after Eid ul-Fitr; it is permissible to start fasting them one or more days after Eid, and they may be done consecutively or separately during the month of Shawal, according to what is easier for a person. There is plenty of room for leniency in this matter, and this is not obligatory, it is Sunnah.
And Allah is the Source of strength. May Allah bless our Prophet Muhammad (Peace and Blessing upon Him) and his family and companions and grant them peace.