Expressing “to ask” with 询问 and 打听
These two verbs have the same meaning “to ask” but are used in different contexts.
The verb 询问
We use the verb 询问 
when we ask questions in a formal setting or in any other situation that requires formalities.
when we ask questions in a formal setting or in any other situation that requires formalities.Examples
你想询问什么？
What do you want to ask about?
nǐ xiǎng xúnwèn shénme
警察询问过路的人是否目击了这次事故
Police asked passersby if they had witnessed the accident
jǐngchá xúnwèn guò lù de rén shìfǒu mùjí le zhè cì shìgù
我打电话询问了那个地区所有的旅行社
I called every travel agency in the area
wǒ dǎ diànhuà xúnwèn le nà ge dìqū suǒyǒu de lǚxíngshè
Sometimes 询问 may be translated as “to inquire”.
Examples
询问此事的信件一律转到新闻处
All correspondence inquiring about this matter is forwarded to the press office
xúnwèn cǐ shì de xìnjiàn yí lǜ zhuǎn dào xīnwén chù
We also can use 询问 as a noun.
Examples
所有询问都将从速处理
All inquiries will be handled promptly
suǒyǒu xúnwèn dōu jiāng cóngsù chǔlǐ
The verb 打听
We use the verb 打听 when we want to ask questions about some information, news, or gossip.
when we want to ask questions about some information, news, or gossip.Examples
记者经常纠缠着打听她的私生活
Journalists often stalk her to find out about her private life
jìzhě jīngcháng jiūchánzhe dǎtīng tā de sīshēnghuó
他到医院去打听她的情况
He went to the hospital to inquire about her
tā dào yīyuàn qù dǎtīng tā de qíngkuàng
我不知道公司有没有空缺，不过我会打听打听
I don't know if there are any vacancies in the company, but I will ask around
wǒ bù zhīdao gōngsī yǒu méi yǒu kòngquē, búguò wǒ huì dǎtīng dǎtīng