Q:

Can you use -ㄹ려고 instead of -려고?

Recently, a Korean woman insisted to me that these are correct sentences:

한국어를 배울려고 했어요 (I intended to learn Korean)
  카페에 갈려고 했어요 (I intended to go to a cafe)

As far as I learned, the grammar is
-려고 하다 intend to do something, so it should be
배우려고 and 가려고. 
Is this a grammar form I am not aware of, or a non-standard use of the language?

A:

If you search "ㄹ려고" in Naver, the 국어사전 (i.e. the Korean to Korean dictionary) says the following:

‘-려고’의 잘못.

In other words, "ㄹ려고" is a mistaken (i.e. incorrect) form of "-려고"
The ensuing discussion on Naver then includes a discussion on the topic. Someone asks the question

일상적으로 말할 때 "이제 막 먹을려고 했다."와 같이 말하는 경우가 있는데 '먹을려고'도 맞는 말입니까?
Translated: "Commonly, when I am speaking, I say something like "이제 막 먹을려고 했다." [I was just about to eat].  Is this correct?

So this person is asking if his or her use of "ㄹ려고" is correct.
The reply is as follows:

아닙니다. '먹으려고'가 표준어입니다. 많은 사람들이 "집에 갈려고 한다."와 같이 말한다고 해서 '-ㄹ려고'를 인정해야 하는 것은 아닙니다. '-ㄹ려고'를 인정할 경우 '가려느냐', '가려다가', '가려더니', '가려면'과 같이 '-려고 하-'가 들어 있는 말들 또한 '갈려느냐', '갈려다가', '갈려더니', '갈려면' 등과 써야 하는데 이러한 말들은 실제로는 잘 쓰이지 않은 어색한 표현입니다. 그러므로 '갈려고'는 잘못임을 알 수 있습니다.
Summarized: No. This is not correct.  '먹으려고' is the correct (표준어) form. Just because people say things like "집에 갈려고 한다" does not mean we need to recognize it as proper. The answering person then goes on to give examples of how the form is commonly misused. But just because people say it, it does not make it correct.

So, I would conclude that, although people say it, it does not make it correct in the strict sense of grammatical rigidity.
The balance here is to recognize that "-ㄹ려고 한다" is a form that is colloquially used. It is not "correct" grammar. But people still use it.  Think of English for a moment. There are all sorts of colloquial misuses of grammar and vocabulary. It would be like someone telling a foreigner that "all y'alls" is the plural of "y'all." People use those phrases, but they are not exactly "proper grammar." Even if a native speaker says it is proper.
My suggestion is that you just stick to using -려고. It is, after all, correct. Sure, some Koreans use slightly bad grammar. Every language has that. But this is one of those times where using proper grammar will actually still sound okay.
Also be aware that sometimes only native speakers can get away with colloquial expressions. Non-native speakers using colloquial grammar or phrases often sounds weird to native ears.
There is one case that I should further elucidate, since it has not been properly addressed yet on this site (even at the question on 쓸려고 versus 쓰려고). When your verb stem has a ㄹ 받침, (so, for example, 알다, to know), the ㄹ 받침 is not dropped from the verb stem (as is common in some other verb endings). The 려고 ending is then directly appended to the verb, without the usual 으 spacer used after all other verb stems ending with a (non-ㄹ) consonant 받침.
Even this though is not a case of ㄹ 려고 하다 being used. It is just a special rule of not dropping a ㄹ 받침.

알다 -> 알려고 하다
먹다 -> 먹으려고 하다

For posterity, here is the Naver link I reference. I have summarized above the most relevant parts.

