This work is devoted to the problems associated with the urgent problem of ensuring asteroid safety. It is about identifying really dangerous objects among the many known ones, finding their possible collisions and approaches to the Earth, as well as the Moon and other planets, and assessing the likelihood of these events. Usually, asteroids dangerous for the Earth are sought among those whose perihelion distances are less than 1.3 AU. («Near-Earth asteroids»), and this seems natural. However, the orbits of asteroids evolve, especially strongly during close encounters with planets. As a result, a transition to the class of «Near-Earth Asteroids» for an object, which was not in this class, is possible, and even its transformation into a really dangerous for the Earth. An assessment of the possibility of such a transformation of the orbit is carried out using the approximate analytical method of point-like gravitational spheres and serious simplifications of the motion model. For this, the plane circular restricted three-body problem «Sun-planet-asteroid» is considered. Interaction with the planet is reduced to an instantaneous rotation of the planetocentric velocity vector, the modulus of which is constant. The elements of the heliocentric orbit, as well as the aphelion and perihelion distances, are transformed according to the formulas of the two-body problem. Using this method, it is easy to show the fundamental possibility of the transition of an asteroid with a perihelion distance close to the radius of Jupiter's orbit to the vicinity of the Earth's orbit. It is known that a close encounter with Jupiter can significantly change the orbit of an asteroid, the approach with Mars is much less. In order to assess the reality of the transition of asteroids to the class of «Near-Earth Asteroids», we performed a numerical integration of a large number of known asteroids, including those with a perihelion distance of more than 1.3 AU. Sequentially selected asteroids that are close to the planets. The calculations were carried out using a recently created high-speed software package (Balyaev I.A. Acceleration of Numerical Integration of the Equations of Motion of Asteroids. Solar System Research, 2020, vol. 54, no. 6, pp. 557–566.) Using the Monte Carlo method, estimates were obtained for the cumulative probabilities of collisions and approaches to planets (including the Earth) for more than 3000 asteroids. The time interval is up to 2132. Each asteroid is modeled by two hundred thousand individual specimens. We found two dozen asteroids with a perihelion distance of more than 1.3 AU, having in a specified time approach to the Earth at a distance of less than 100 of its radii. For almost all of these asteroids, encounters and possible collisions with Jupiter have been recorded, and many approaches have been recorded with Mars. Two asteroids are approaching Venus by less than 100 of its radii, two asteroids are approaching Saturn by less than 100 of its radii. Consider asteroid 2011 XD3 as an example. Its perihelion distance is 1.53 AU, accuracy is 0.0008 AU (1 sigma), aphelion distance is 5.16 AU, accuracy is 0.07AU (1 sigma). Two approaches of this asteroid to the Earth at less than 100 of its radii were recorded, 6 approaches to Mars at less than 100 of its radii, 3751 approaches to Jupiter at less than 100 of its radii, 512 approaches to Jupiter at less than 10 of its radii, 103 possible collisions with Jupiter. Summing up, we can say that asteroids that do not yet belong to the number of «Near-Earth Asteroids» can pose a danger to the Earth. Isolation and study of such objects is an urgent task, its solution is possible with the use of modern computing facilities.