Three subjects are expected to dominate the summit: European security, arms control and efforts to improve the Russian economy.
But it is Nato's determination to expand into central and eastern Europe that is likely to provoke the most emotional exchanges, leaving neither leader fully convinced that the other genuinely understands his position.
For some months, Mr Yeltsin has recognised the inevitability of Nato expansion and has concentrated on securing guarantees that the process will not endanger or isolate Russia.
His latest demand, made last week, is an assurance that Nato will not take in former Soviet republics - specifically, the Baltic states and Ukraine.
Publicly, it will prove difficult for Mr Clinton to accept this demand, since the Balts and Ukrainians would be outraged at any suggestion that the United States was implicitly consigning them to a Russian sphere of influence. Privately, however, Mr Clinton should be able to give Mr Yeltsin some words of comfort.
First, Nato has no immediate plans to incorporate the Baltic states and will have its hands full for years with the business of absorbing the Czech Republic, Hungary and Poland - the three leading candidates for membership in 1999. Indeed, the US believes that the best scenario for the Baltic States would be inclusion in the first wave of European Union enlargement - although some EU countries beg to differ on that point.
Second, Nato's strategy for Ukraine is not to make it a full member but to establish a special relationship of the kind the alliance is trying to forge with Russia. There is, as Mr Yeltsin knows, no question of Nato basing nuclear weapons or alliance troops on the territory of new member states, let alone the Baltic republics or Ukraine.
However, Mr Yeltsin may object that Russia believes it won a pledge from the US and Germany that, in return for German unification in 1990, Nato would not expand at all into Eastern Europe.
For that reason, Russia this time wants a legally binding treaty restricting the terms of Nato enlargement.
Although the US rules out such a commitment, the summit may produce a compromise. The treaty defining Nato's relationship with Russia would be approved by all the member states' parliaments and would be as "politically binding" as was the 1975 Helsinki Final Act on European security and co- operation.
Even a deal along these lines may not be enough to banish the Russian suspicion that Nato is taking advantage of Russia's temporary weakness to absorb a part of Europe from which Russia has been invaded many times in the past.
However, Mr Clinton is likely to stress that Nato sees Russia as a long- term partner and that, in any case, Russia's main security challenges in the coming century will not come from the West but from its southern Islamic neighbours and China.
To convince Mr Yeltsin of his good faith, Mr Clinton is expected to announce support for Russian entry into important clubs such as the World Trade Organisation and the Organisation for Economic Co-operation & Development (OECD). He may also hold out the prospect of upgraded status for Russia at annual meetings of the G-7 group of leading industrial countries.
In return, however, Mr Yeltsin will have to promise a renewed commitment to economic reform, including the elimination of cor- ruption, the stabilisation of the government budget, and the creation of proper legal (not to mention physical) conditions for foreign investors and businessmen.
The appointment of Anatoly Chubais and Boris Nemtsov to mastermind the latest phase of reform may have reassured Mr Clinton that Russia is back on course after a year of drift. The prospects for progress on nuclear arms control at the summit are uncertain. Mr Yeltsin could promise a new effort to persuade the Russian parliament to ratify the 1993 Start-2 Treaty, which cuts the US and Russian nuclear arsenals by about two-thirds.
But the parliament, dominated by Communists and Nationalists, is in no mood to approve Start-2 as long as Nato insists on expansion. Since the US says that negotiations on a Start-3 Treaty cannot begin until the parliament ratifies Start-2, it is unclear whether much progress can be made in Helsinki.
However, neither leader has an interest in seeing the summit fail. There will be hard bargaining in Helsinki, especially over Nato enlargement, but on balance the summit is likely to warm up US-Russian relations rather than cast them into deep freeze.