From the motion:

We seek this information based on our belief that these contacts were among the precipitating events for Tamerlan’s actions during the week of April 15, 2014, and thus material to the defense case in mitigation. We base this on information from our client’s family and other sources that the FBI made more than one visit to talk with Anzor, Zubeidat and Tamerlan, questioned Tamerlan about his internet searches, and asked him to be an informant, reporting on the Chechen and Muslim community.

We further have reason to believe that Tamerlan misinterpreted the visits and discussions with the FBI as pressure and that they amounted to a stressor that increased his paranoia and distress. We do not suggest that these contacts are to be blamed and have no evidence to suggest that they were improper, but rather view them as an important part of the story of Tamerlan’s decline. Since Tamerlan is dead, the government is the source of corroboration that these visits did in fact occur and of what was said during them.

The FBI acknowledged briefly investigating Tamerlan in this April, 2013 press release:

Once the FBI learned the identities of the two brothers today, the FBI reviewed its records and determined that in early 2011, a foreign government asked the FBI for information about Tamerlan Tsarnaev. The request stated that it was based on information that he was a follower of radical Islam and a strong believer, and that he had changed drastically since 2010 as he prepared to leave the United States for travel to the country’s region to join unspecified underground groups.

In response to this 2011 request, the FBI checked U.S. government databases and other information to look for such things as derogatory telephone communications, possible use of online sites associated with the promotion of radical activity, associations with other persons of interest, travel history and plans, and education history. The FBI also interviewed Tamerlan Tsarnaev and family members. The FBI did not find any terrorism activity, domestic or foreign, and those results were provided to the foreign government in the summer of 2011. The FBI requested but did not receive more specific or additional information from the foreign government.

Here's a Wall St. Journal report of his father's account:

Earlier, Tamerlan had attracted the attention of the FBI. His father said the officers visited one-and-a-half years ago to discuss Tamerlan's interests and had tea in his third-floor apartment. "They told me they were watching everything—what we look at on the computers, what we talked about on the phone," he said. "I said that's fine. That's what they should be doing."

The boys' mother, Mrs. Tsarnaeva, said Tamerlan was defiant in his meetings with FBI officials. "I am in a country that gives me the right to read whatever I want and watch whatever I want,'" Mrs. Tsarnaeva recalled her son saying. "He was even trying to get the FBI [agent] to convert to Islam."

There are also media reports like this one in the New York Times that Tamerlan and Dzhokhar's mother's phone calls were being monitored by Russia and she, rather than Tamerlan was the focus of the inquiry.
The article also reports his mother said he was stopped by police for no reason and very upset:

It is unclear how closely the police were tracking Mr. Tsarnaev, but his mother described at least one instance in which her son was stopped by the police along the beach in Makhachkala, where Mr. Tsarnaev’s parents live, and brought in for questioning.

“He’s like: ‘The police came there and they asked for documents,’ ” Ms. Tsarnaeva said at a news conference last week. “They asked him to follow. He was asking them, he was like in shock. He’s like: ‘What, is there something wrong with me? Am I strange, or don’t look like everybody?’ ”

In October, 2013, the FBI denied it tried to recruit Tamerlan:

“The Tsarnaev brothers were never sources for the FBI, nor did the FBI attempt to recruit them as sources.”