Can たがる + そうです?

Question

As far as my knowledge, when we talk about the third person's desire and feeling, “Someone wants to do something” , we'll use 〜たがる instead of 〜たい - your own desire and feeling or when you ask someone about their desire.

For example:

田中さん、旅行したいそうですね!
Hey Tanaka, I heard that you want to travel!
(I'm talking with Tanaka)

So is it possible to be like:

田中さんは旅行したがるそうです.
I heard that Tanaka wants to travel.

Answer 1

Grammatically, that seems just fine -- ～たがる to express that someone else wants to do something, and そうだ to express reported speech sourced from someone else.

Google can sometimes be helpful in confirming if a particular structure is in common use. Searching for "たがるそう" shows plenty of hits. Some of those hits are not relevant, such as when there's a period in between -- 「たがる。そう」 does not demonstrate what we're looking for. However, the hits do include plenty of fitting examples, such as: 「僕の彼女のお父さんは、同じ場所ばかり行きたがるそうです。」 ("Apparently my girlfriend's father only ever wants to go to the same places.")

Answer 2

They mean different things.

田中さんは旅行したいそうです means that you heard of Tanaka-san's current desire to travel around, while 田中さんは旅行したがるそうです is saying that you heard of his habit to abruptly try to go on a trip from time to time whether he currently wants or not.

Answer 3

Can たがる + そうです?
As far as my knowledge, when we talk about the third person's desire and feeling, “Someone wants to do something” , we'll use 〜たがる instead of 〜たい - your own desire and feeling or when you ask someone about their desire.

No, you probably want to avoid using it for other than family members or very close friends because otherwise it could sound disrespectful.

た of たがる is of たい(want).
がる has a complicated meaning to say someone/something indicates/shows their needs/wants consciously or not. It has a sense of persistency, and can be also used to say an intentional display, as 欲しがる or 寒がる comparing to 欲しそう or 寒そう. We use this for the first person as well as from the perspective of others seeing it; 私が食べたがっているのに一口もくれない。

たがる
［助動］
《希望の助動詞「たし」の語幹「た」＋接尾語「がる」から》

１ 話し手以外の人の希望を表す。「この子はお菓子を食べたがってしかたがない」
２ （「たがっても」の形で）話し手の希望を表す。「私がどんなに退院したがっても、医者が許してくれない」
(デジタル大辞泉|たがる)

So, したがるそうです is used as a reported speech about someone under someone's care; 患者{かんじゃ}さんが外へ出たがるそうです.