For the three kingdoms 190-284 CE, no bar one occurance. I'm also unaware of duels during the later Han dynasty wars either.
I can think of one example of a prearranged duel:
On the 22nd of May 192 the tyrannical controller of the Han Dong Zhuo, having survived a few plots, was killed by his bodyguard Lu Bu and his chief minister Wang Yun. They quickly destroyed Dong Zhuo's family at the capital and his stronghold at Mei. However some of Dong Zhuo's officers had been away raiding during this regime change and returned home to an uncertain situation, they sought an amnesty as an assurance of safety but Wang Yun refused. His execution of the famed scholar Cai Yong and high-handed manner alienated people while there were rumours that Wang Yun was going to kill figures from Liang province (the home of those returning generals) So, on the advice of Jia Xu, they allied with each other with Li Jue treated as the senior general) and marched on the capital Chang'an with the armies of the new regime deserting or being defeated.
When Li Jue and party reached the capital, according to the scholar Wang Can who was at the capital at the time, there would be a duel between senior general Guo Si and Lu Bu, a man famed for his abilities as a warrior. Wang Can's account is an annotation in Lu Bu's SGZ, translation Yang Zhengyuan
Guō Sì was north of the city. Bù opened the city gates, led troops to face [Guō] Sì, and said: “Let us both withdraw troops, and ourselves personally decide victory.” [Guō] Sì and Bù then dueled, and Bù with a spear stabbed [Guō] Sì. Riders behind [Guō] Sì then came forward to rescue [Guō] Sì. [Guō] Sì and Bù then each withdrew.
Lu Bu won the duel but would lose the war as he was forced to flee Chang'an amidst a mutiny, the new regime lasting less then sixty days from killing Dong Zhuo. There were very rare other duels (Sun Ce vs Taishi Ci, Ma Chao vs Yan Xing), but a case of people coming across each other rather then prior to battle.
The era is known for duels becuase of entertainment like Romance of the Three Kingdoms, a novel written over a thousand years later but which heavily influences the way people see the era. The way battles were fought in the era was changed, no longer focus on logistics, limited control and trying to make break through via disrupting the enemy cohesion. One of the changes is that so many battles open with, or quickly begin, with duels including the opening battle of the novel. Skills of bravery and martial might, the loser (very quickly or after several bouts) possibly being killed and the victor's army quickly pushes on to take advantage, win the duel and win the day. There are memorable feats like the ageing Zhao Yun killing the (fictional) Han family of a father and seven sons in a northern camapign, Guan Yu fighting his way through the five passes, Lu Bu the mighty facing the three brothers at Hulao Gate, all fictional (and two of them in fictional camapigns).
The thrill of great warriors testing their mettle in displays of great valour and ability excites the readers and viewers of modern day adaptation like Red Cliff. However the novel's style of warfare had very little to do with the historical style of warfare of the era.
Sources
Records of the Three Kingdoms by Chen Shou, annotations by Pei Songzhi and translation by Yang Zhengyuan
Imperial Warlord: A Biography of Cao Cao 155-220 AD by Rafe De Crespigny. Chapter 4 particularly, the conduct of the civil war, goes into the way camapigns and battles were fought.
Generally those are quite rare as we go into the medieval periods, and certainly not as common as portrayed in media. As u/Dongzhou3kingdoms mentioned, entertainment products like novels and films often focus more on the actions and individual heroism and ideals, rather than more historically-based considerations.
Other than the case of Guo Si and Lu Bu, I could also think of a case of attempted arrangement of a duel, between Liu Bang and Xiang Yu during the Chu-Han Contention period.
In the year 203 BC, Liu Bang and Xiang Yu clashed near Guangwu. Here they had a long stand-off. This was when Xiang Yu requested a duel with Liu Bang.
He said:
天下匈匈數歲者，徒以吾兩人耳，願與漢王挑戰決雌雄，毋徒苦天下之民父子為也。
For the past few years, the world has been turbulent and chaotic only because of the two of us. I wish to challenge you to a duel and fight it out, so as not to make the people suffer anymore.
Liu Bang declined:
吾寧鬬智，不能鬬力。
I would rather have a battle of wits than a battle of might
Xiang Yu then ordered brave warriors to go out and challenge Liu Bang's soldiers. But all the three who went were killed. Xiang Yu was furious and personally went in to challenge the soldier. The soldier was frightened and retreated. Liu Bang was also surprised that Xiang Yu personally came forward.
Then they had a chat across the Guangwu canyon, and Xiang Yu challenged Liu Bang to a duel again. Liu Bang declined, again. He also accused Xiang Yu of committing horrendous sins. Xiang Yu was furious and Liu Bang was also injured by hidden bows. And with that, ends the failed attempts by Xiang Yu to arrange a duel with Liu Bang.
Records of the Grand Historian by Sima Qian
Edit:
Upon some further thinking I realize this might have something to do with a popular pre-Qin way of war called 致師 (zhì shī).
Eastern Han scholar Zheng Xuan in his annotations of Rites of Zhou said:
致師者，致其必戰之志。古者將戰，先使勇力之士犯敵焉。
zhì shī is a demonstration of morale. When the ancients battle, [there shall] first be brave warriors dispatched to fight the enemy.
These brave warriors need not be champions or famous generals, and they need not be in individuals. But there seems to be a certain resemblance to the sort of pre-battle duels OP asked about, and possibly had a certain influence on later works/fights, because Yuchi Gong seemed to have 'called' for it as well during one of his fights.
From New Book of Tang:
王命李勣等為伏，親挾弓，令敬德執槊，略其壘，大呼致師。
[Li Shiming] ordered Li Ji and co to ambush, while personally holding a bow. He also ordered Yuchi Jingde, with his lance in hands, to request for zhì shī loudly in front of [Dou Jiande's] camp.
Unfortunately I'm not familiar with ancient China during the pre-Qin periods, so hopefully someone else more knowledgeable in this can carry this further.
To build on the answers given by u/Dongzhou3kingdoms and u/10thousand_stars, I believe part of the depiction of such duels in the romantic depictions of the histories does result from an expectation that officers were to lead from the front, at least in certain periods of history. I have an interest in the Three Kingdoms period of China, so I know a little bit about that. u/Dongzhou3kingdoms has already referenced Rafe de Crespigny's Imperial Warlord, and "Chapter 4: The Conduct of Civil War" has this to say on the nature of battle conducted in that time period:
"Though the formal structure of military units echoed that of Han, with generals and colonels and lesser ranks identified in the texts, such hierarchy in the order of battle conceals the reality of rivalry, conflict and confusion. We have seen how armies were organised in bands led by individual fighters, but the very nature of such warfare meant that leaders were not only skilled in hand-to-hand combat and personally charismatic; they were also aggressive, over-confident, often prone to violence, and sometimes a little mad: what sort of man could face a mass of opponents without certainty that he would be successful, or even supported?"
Maybe de Crespigny is over-egging a bit here, but there are many moments in the Three Kingdoms era where famous warriors personally perform acts of violent derring-do. For example, Guan Yu famously slays Yan Liang during the Guandu campaign, singling him out amongst his forces and returning with Liang's head to Cao Cao. And while Guan Yu's deed is quite exceptional, he's by no means alone.
The Sanguozhi reports that Ling Tong, an officer of Sun Quan, personally led but tens of men ahead of his main forces in Sun Quan's campaigns in Jiangxia (208), beheading an enemy general Zhang Shuo and capturing his mariners.
Another example would be Zhang Liao, a general of Cao Cao, who during the defense of He Fei (214) against Sun Quan's forces opened the battle by leading a small sortieing force into enemy lines. In the SGZ, he is reported to have personally slain tens of men and two generals, and broke through enemy lines twice in order to rescue his encircled forces.
Ling Tong actually shows up again here, with another example of personal participation in combat: in the close of this failed campaign, Sun Quan decided to have his forces retreat, but remained behind with a small number of soldiers. Zhang Liao seized this opportunity to himself encircle Sun Quan. It was only due to the intervention of Ling Tong, who in this battle received many wounds and suffered the total loss of his three hundred personal troops, that Sun Quan managed to escape.
You can find many such incidents peppered throughout the biographies of eminent individuals during the Three Kingdoms era.
So while I would agree that set-piece duels that prelude battles are rare, at least in the Three Kingdoms era it wasn't uncommon for renowned officers to engage personally in combat, as well as against one another in certain contexts.