Politically, General Secretary Zhao Ziyang wanted to resolve the situation peacefully by negotiating with the demonstrators. He made this known publicly during his May 4th speech to the Asian Development Bank, where he defended the protesters and conveyed the need to democratically and coolly resolve the situation according to law.

When protests continued to escalate, Zhao asked Deng Xiaoping to hold an one on one meeting to discuss how to resolve the crisis. Zhao was surprised to see other members of the politburo there. Zhao was further surprised when two politburo members (Li Peng and Yao Yiling) blamed his May 4th speech as the cause of the escalation of protests.

This surprise attack eroded his power at the meeting and helped gain enough support for the imposition of martial law. Zhao basically lost all power after this meeting and eventually was kept under house arrest for the remainder of his life. He was also basically erased from Chinese history.

Zhao himself blamed the April 26th editorial in the People's Daily for the escalation of protests. In the editorial, the demonstrations were condemned as "anti party, anti socialist turmoil". Zhao was trying to convince Deng Xiaoping to only punish those who beat, smash, loot, burn or trespass. Later, he also tried to convince Deng to revise the April 26th editorial to appease the protesters

In both the politburo meeting and the publishing of the editorial, Zhao was caught off guard. Zhao claimed that Deng and others agreed with his position. However, in both instances people talked behind his back and gained support for taking a hard line approach to the demonstrations. This secrecy seems to indicate that Zhao still had some support in the government, otherwise they would have just openly turned him down at the start.

Also, in a letter Zhao wrote to Deng trying to get him to reconsider the imposition of martial law, Zhao wrote that many "prominent people and senior party members" (such as Li Yimang) had implored him to acknowledge that "the students actions were patriotic, and to change the wrongful stance toward the students." So it would seem that Zhao was not the only person in the government showing reticence. Rather, he was just the highest ranking one.

All of this is according to the book "Prisoner of the State". The book was created through tape recordings that Zhao secretly made during his house arrest into one of his granddaughter's toys which had a tape recording function. While it only presents one point of view of the inner workings of the government during that time, it also represents one of the only (or maybe THE only) points of view since the inner workings of the politburo are secret.

Edit: April 26th editorial, not April 27th.

Actually a fair number of military leaders opposed the usage of soldiers to crack down on the protesters. This fact tends to get ignored because everyone will focus on Zhao Ziyang when talking about opposition to the Party leaders. The Wall Street Journal reported in 1991 that as many as 3,500 PLA officers disobeyed orders to suppress the protesters. One of the most famous cases, which the New York Times covered a few years back, was the case of General Xu Qinxian. Xu commanded the elite 38th Army Group (now renamed the 82nd Group Army), one of the three armies of the Beijing Military Area that protected the capital. Xu refused to follow orders to enforce martial law, stating that he would "rather be beheaded than become a person condemned by history." He was concerned that using the military to suppress the student movement would stain the reputation of the PLA. He ended up giving the order to his subordinates, but refused to lead his troops in person. Xu was court-martialled, sentenced to five years in prison, and expelled from the party.

There were also other cases of resistance. The 39th Army Group's 116th Division purposely ignored orders to advance into Tiananmen Square by pretending its radios had broken down. Its commander, Xu Feng, was demoted as a result. The 28th Group Army also did not enforce martial law. Its commander General He Yanran and Political Commissar Zhang Mingchun refused to carry out orders to forcibly dispel the protesters. Instead, soldiers of the 28th Group Army deserted en masse. Most of its equipment was left behind and burned by the protesters. He Yanran, Zhang Mingchun, and Chief of Staff Qin Jinkai were court-martialled and demoted. The entire army was ordered to undergo six months of reorganization and all its commanding officers were reassigned.

Even the highest-ranking generals in the PLA were against sending in the PLA. A petition was circulated and signed by major military figures such as Generals Zhang Aiping, Ye Fei, Xiao Ke, Yang Dezhi, Chen Zaidao, Song Shilun, and Li Jukui, then submitted to the Central Military Commission. The Defense Minister at the time, Qin Jiwei, privately attempted to forge an alliance with Zhao Ziyang to countermand Deng's order for martial law. In fact, the perceived military pressure against the enforcement of martial law was so great that Deng Xiaoping and major CCP figures seriously feared that there would be a military coup.

The Tiananmen Crackdown definitely served as an example to the CCP leadership on how they should deal with protests and riots in the future. The state began to emphasize the usage of the People's Armed Police, the paramilitary police force, which it has used to suppress riots in Tibet and Xinjiang, but that's getting more into the recent stuff.